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Someone just asked me this question, and it is a good question. The dilemma of writing characters smarter than oneself has plagued me before. As I was analyzing this and writing a response, my answer became a whole study paper. So, I am posting it here so others can make use of it as well.

These are only my own observations and terminology, put together on the fly. There might be a few holes here, but hopefully it will serve to generate ideas on how you can create the characters you want.

First, there are different types of "smart". There are a number of talents often associated with smart characters, and they each have their own uses. Some characters have only one of these. Some have several. If you want to make a ridiculous, over-powered character . . . I dare you to roll them all into one!

Book Smarts:

Examples: Hermione from Harry Potter, or Encyclopedia Brown.

Description: A character with this talent is a walking Wikipedia/Dictionary/etc. They know lots of information, but they aren't necessarily intelligent. Sometimes they just have really good memories.

How to Write One: Have some idea of the information the character is going to need to accomplish their objectives. Study it yourself. If the guy is lost in the wilderness at night and needs to find direction, this could be a number of things: star maps, the direction the local wind usually blows at night, how to make a needle compass, the migratory patterns of local animals, local landmarks--whatever.

Plus, studying things your character knows will give you ideas and help you add details. This helps you avoid something vague and boring: "He checked the stars to find which way was north." Instead, you can make it more interesting: "He stared at the stars. Which way was north? The only constellation he knew was the Big Dipper, and he couldn't see it anywhere. No. No, wait. Was that the tail of it creeping over the distance mountains? Yes! That was north!"

For a comic, it would be more along the lines of the character saying, "Ah-ha! There's the Big Dipper, so north is this way."

Observation:

Examples: Sherlock Holmes, the guy in Lie to Me, and many other detective characters--also, trackers, hunters, and assassins

Description: A character with this talent notices everything, down to the smallest details.

How to Write One: Know what they need to notice to accomplish their objectives, but don't be afraid to string them along. Maybe noticing a bit of unimportant lint on the floor leads them to notice a scrape mark left by a hidden door. These characters are pretty easy to write. All it is, is stringing breadcrumbs.

Deduction:

Examples: Sherlock Holmes, Patrick Jane from The Mentalist, Sheriff Jack Carter from Eureka, and many other detective characters

Description: A character with this talent draws conclusions from information, and is usually paired with Observation or Book Smarts. These complementary talents can be placed in another character, or all in one character.

How to Write One: Work backwards. What conclusion does the character need to draw? Now figure out a way to lead him to it. He needs to figure out there is a trap in the hall ahead? What is the trap? Is it C-4 explosive? C-4 smells like almonds (or so I hear), so maybe the character smells almonds and concludes C-4 is involved and there might be a trap. Might is the keyword, though. To make it believable (unlike The Mentalist), you need to consider that the smell of almonds could actually be . . . almonds (on the breath of a hiding assassin? in the pack of a companion?). Or, it could be C-4 and not be a trap at all (an nearby weapons cache? a hiding demolitions expert?).

Awareness:

Examples: Ninjas, leaders, and guys with all the connections

Description: This talent is similar to Observation, but it is different. An observant person might notice a gold coin on the floor and pick it up. An aware person will notice the man in the shadows who is about to ambush the guy picking up the coin. Awareness can relate to more than immediate surroundings--a mob boss or a CIA agent might have so many connections to sources and snitches, that they know everything that is going on in a town or a country. A good military officer is aware of where each of his soldiers are, what they are doing, and how they are doing. This is often a talent given to leader characters or evil masterminds.

How to Write One: For ninja-type awareness, you're looking at a person who almost has a sixth sense, or "eyes in the back of their head". Simply be aware of what is going on it the character's surroundings and make them aware of it, too. Is someone sneaking up on them? Is the mob turning ugly? Did all the forest creatures suddenly fall quiet? This can be pointed out in specifics, like hearing a footstep, or in generalities: "Something didn't feel right." These characters keep track of people around them, so if they are in a house, they probably know who is in the bathroom, who is working in the garage, and who is in the kitchen--even if all of these people aren't currently in eyesight.
For connection-type awareness, keep in mind where the character is getting their information. Is it from a radio call from the front lines? A rumor reported by a snitch in a back alley? A spy inside enemy lines? A newspaper? A bugged phone line? The hardest part may be getting a source into position. How does this character know what went on in the enemy's secret meeting? Does he have a spy? A bug in the room? A hearing amplifying tool or spell? Is the how even important? Sometimes you can make a villain more unnerving if no one can figure out how he knows everything (but this can make it less believable if you don't leave a few clues).

Intelligence:

Examples: Henry Deacon or Allison Blake (or most of the town) from Eureka, computerized characters

Description: This talent is similar to deduction, but this is a person who can grasp a situation and process it quickly. They are adaptable, because intelligence gives an edge in any situation, whether it is foreign to the character or not. Give them a problem relating to racehorses and they'll offer solutions just as valid as if you asked about car engines or math. If brains are computers, this character has a powerful CPU.

How to Write One: Writing a character more intelligent than oneself can be tricky. In fact, it almost sounds preposterous. But you, as the writer, have the advantage of time. A computer with a slow CPU can do the same calculations as the computer with a powerful CPU--it just takes longer. If you're like me, a math problem might fluster you, but that doesn't mean you can't solve it if you sit down and take your time. You might even have to look up how to solve the problem. But once you have the answer, your character can produce the answer mere seconds after being given the problem. Making the character come to conclusions rapidly and without doubt will make them seem smarter. Make them right every time. Make them correct other characters' faulty logic. Intelligent people may also get impatient or arrogant around people who aren't as smart.

Deviousness:

Examples: Rube Goldberg, the Weasley twins from Harry Potter, evil masterminds

Description: A character with this talent is not necessarily knowledgeable or intelligent, or anything else. Their skill is the ability to think outside the box, to come up with crazy solutions that sometimes even work. Trickery and manipulation are also tools of the devious person.

How to Write One: These might be the hardest characters to write. You have to think like them. You have to ask yourself, how would someone solve this problem? And then after you get through all the standard approaches, you find the clever ones. Trying to not think in a straight, logical path can help. Think of the story of Icarus. He and his father were imprisoned (in a tower, I think?). Standard methods of escape are climbing down, overpowering a guard, etc. But no! This genius builds WINGS out of feathers and wax. Brilliant? Or just insane? Either way, it worked. Mostly. Other devious ways to escape could have included playing mind games with the guards or the captor. Manipulation of people is a classic "devious" trait. Devious people do not "break up" with their boyfriends--they drive them away with undesirable habits, excessive clingy-ness, nagging, or even by hiring another girl to woo the boy away. Rather than divorce a wife and have to split the estate, why not slowly drive her to insanity? Or have her kidnapped? Or so forth. It doesn't have to be sadistic. Good people can be devious, too.

The great thing about deviousness is that it is more colorful and unpredictable that simple intelligence. It also lends itself to humor. And, dumb characters can be devious, too, in their own way (even unintentionally), so this can level a playing field.

Telepathy:

Examples: Professor Xavier and Jean Grey from X-Men, Matt Parkman from Heroes, Jedi/Sith

Description: This talent is gives the character the ability to read minds and sometimes plant thoughts. It has little to do with actual intelligence, but it gives the character the ability to know anything any other character knows. This can give the illusion of all the above skills combined.

How to Write One: This talent usually gets relegated to super heroes, magical worlds, supernatural worlds, etc. However, it is something that has been extensively researched by assorted militaries, and the idea of telepathy is filtering into popular "realistic" science fiction as well (StarCraft, Firefly). Be aware of your genre and what your readers will or won't reject.

Since telepaths can read minds, they can pull any facts, observations, deductions, ideas, etc. from other characters. This makes them functionally almost as smart as anyone else. Telepaths see things coming and prepare accordingly. They can read emotions, so they are exceptional at mind games and manipulation whether or not they can do "Jedi mind tricks".

There are a lot of things to consider when writing a telepath, like how to not make them completely over-powered. A person who can read and "write" minds will need "Kryptonite". They need a weakness, or their opponents need some sort of shield or counter attack.

Also, what is the extent of their skills? Consider the following:

-Gut instinct (For the purpose of writing, this is like the bottom of the barrel telepathy. This is the cop who can tell what people are trustworthy, the detective who knows when people are lying, the soldier who gets a bad feeling before an ambush.)

-Empathy (This would be a person who can tell when someone is nervous or happy, even if they can't see the person. They can feel what the other person feels.)

-Intuition/Low-Level Mind Reading (Here we have the slight ability to predict others. You see this in people who finish each other's sentences. You can say, "Hey, you remember the thing on the thing that one time we went to that thing?" And they know exactly what you mean. Sometimes.)

-Medium-Level Mind Reading (This would be the person who can read exact thoughts at whim. If you're telling them about a cat and you're thinking about a dog, they would be able to tell you what breed of dog, what color, etc.)

-High-Level Mind Reading (This is the person who can dig into not just your thoughts, but your memories. Stuff you aren't even thinking of. No secret is safe!)

-Manipulator (The Jedi mind trick. This person can influence the behavior of the weak-minded. If they are powerful, maybe even the strong-minded.)

-Implanter (By far the scariest character, this person can erase and replace your memories, leaving you with no way to be certain that you know what you're doing, why you are doing it, or even who you are.)

Additional limitations:

-Range (Do they have to be touching their "victim"? With ten feet? Anywhere in the world? Line of sight?)

-Taxation (Does their ability drain them mentally or physically? What do they have to trade for their ability? Anything?)

-Mental fortitude (Can others with strong wills fend them off?)

-Physical interference (Does a tin foil hat ward them off? How about Magneto's helmet?)

-Physical tools (Is their power bestowed by a machine or magic item?)

-Nemesis (Is there a particular person they cannot affect, or particular people who have the ability to stop them?)

I hope this reference is helpful. If you have any questions, or think something should be added/revised on here, let me know.

Description
Someone just asked me this question, and it is a good question. The dilemma of writing characters smarter than oneself has plagued me before. As I was analyzing this and writing a response, my answer became a whole study paper. So, I am posting it here so others can make use of it as well.

These are only my own observations and terminology, put together on the fly. There might be a few holes here, but hopefully it will serve to generate ideas on how you can create the characters you want.

First, there are different types of "smart". There are a number of talents often associated with smart characters, and they each have their own uses. Some characters have only one of these. Some have several. If you want to make a ridiculous, over-powered character . . . I dare you to roll them all into one!

Book Smarts:

Examples: Hermione from Harry Potter, or Encyclopedia Brown.

Description: A character with this talent is a walking Wikipedia/Dictionary/etc. They know lots of information, but they aren't necessarily intelligent. Sometimes they just have really good memories.

How to Write One: Have some idea of the information the character is going to need to accomplish their objectives. Study it yourself. If the guy is lost in the wilderness at night and needs to find direction, this could be a number of things: star maps, the direction the local wind usually blows at night, how to make a needle compass, the migratory patterns of local animals, local landmarks--whatever.

Plus, studying things your character knows will give you ideas and help you add details. This helps you avoid something vague and boring: "He checked the stars to find which way was north." Instead, you can make it more interesting: "He stared at the stars. Which way was north? The only constellation he knew was the Big Dipper, and he couldn't see it anywhere. No. No, wait. Was that the tail of it creeping over the distance mountains? Yes! That was north!"

For a comic, it would be more along the lines of the character saying, "Ah-ha! There's the Big Dipper, so north is this way."

Observation:

Examples: Sherlock Holmes, the guy in Lie to Me, and many other detective characters--also, trackers, hunters, and assassins

Description: A character with this talent notices everything, down to the smallest details.

How to Write One: Know what they need to notice to accomplish their objectives, but don't be afraid to string them along. Maybe noticing a bit of unimportant lint on the floor leads them to notice a scrape mark left by a hidden door. These characters are pretty easy to write. All it is, is stringing breadcrumbs.

Deduction:

Examples: Sherlock Holmes, Patrick Jane from The Mentalist, Sheriff Jack Carter from Eureka, and many other detective characters

Description: A character with this talent draws conclusions from information, and is usually paired with Observation or Book Smarts. These complementary talents can be placed in another character, or all in one character.

How to Write One: Work backwards. What conclusion does the character need to draw? Now figure out a way to lead him to it. He needs to figure out there is a trap in the hall ahead? What is the trap? Is it C-4 explosive? C-4 smells like almonds (or so I hear), so maybe the character smells almonds and concludes C-4 is involved and there might be a trap. Might is the keyword, though. To make it believable (unlike The Mentalist), you need to consider that the smell of almonds could actually be . . . almonds (on the breath of a hiding assassin? in the pack of a companion?). Or, it could be C-4 and not be a trap at all (an nearby weapons cache? a hiding demolitions expert?).

Awareness:

Examples: Ninjas, leaders, and guys with all the connections

Description: This talent is similar to Observation, but it is different. An observant person might notice a gold coin on the floor and pick it up. An aware person will notice the man in the shadows who is about to ambush the guy picking up the coin. Awareness can relate to more than immediate surroundings--a mob boss or a CIA agent might have so many connections to sources and snitches, that they know everything that is going on in a town or a country. A good military officer is aware of where each of his soldiers are, what they are doing, and how they are doing. This is often a talent given to leader characters or evil masterminds.

How to Write One: For ninja-type awareness, you're looking at a person who almost has a sixth sense, or "eyes in the back of their head". Simply be aware of what is going on it the character's surroundings and make them aware of it, too. Is someone sneaking up on them? Is the mob turning ugly? Did all the forest creatures suddenly fall quiet? This can be pointed out in specifics, like hearing a footstep, or in generalities: "Something didn't feel right." These characters keep track of people around them, so if they are in a house, they probably know who is in the bathroom, who is working in the garage, and who is in the kitchen--even if all of these people aren't currently in eyesight.
For connection-type awareness, keep in mind where the character is getting their information. Is it from a radio call from the front lines? A rumor reported by a snitch in a back alley? A spy inside enemy lines? A newspaper? A bugged phone line? The hardest part may be getting a source into position. How does this character know what went on in the enemy's secret meeting? Does he have a spy? A bug in the room? A hearing amplifying tool or spell? Is the how even important? Sometimes you can make a villain more unnerving if no one can figure out how he knows everything (but this can make it less believable if you don't leave a few clues).

Intelligence:

Examples: Henry Deacon or Allison Blake (or most of the town) from Eureka, computerized characters

Description: This talent is similar to deduction, but this is a person who can grasp a situation and process it quickly. They are adaptable, because intelligence gives an edge in any situation, whether it is foreign to the character or not. Give them a problem relating to racehorses and they'll offer solutions just as valid as if you asked about car engines or math. If brains are computers, this character has a powerful CPU.

How to Write One: Writing a character more intelligent than oneself can be tricky. In fact, it almost sounds preposterous. But you, as the writer, have the advantage of time. A computer with a slow CPU can do the same calculations as the computer with a powerful CPU--it just takes longer. If you're like me, a math problem might fluster you, but that doesn't mean you can't solve it if you sit down and take your time. You might even have to look up how to solve the problem. But once you have the answer, your character can produce the answer mere seconds after being given the problem. Making the character come to conclusions rapidly and without doubt will make them seem smarter. Make them right every time. Make them correct other characters' faulty logic. Intelligent people may also get impatient or arrogant around people who aren't as smart.

Deviousness:

Examples: Rube Goldberg, the Weasley twins from Harry Potter, evil masterminds

Description: A character with this talent is not necessarily knowledgeable or intelligent, or anything else. Their skill is the ability to think outside the box, to come up with crazy solutions that sometimes even work. Trickery and manipulation are also tools of the devious person.

How to Write One: These might be the hardest characters to write. You have to think like them. You have to ask yourself, how would someone solve this problem? And then after you get through all the standard approaches, you find the clever ones. Trying to not think in a straight, logical path can help. Think of the story of Icarus. He and his father were imprisoned (in a tower, I think?). Standard methods of escape are climbing down, overpowering a guard, etc. But no! This genius builds WINGS out of feathers and wax. Brilliant? Or just insane? Either way, it worked. Mostly. Other devious ways to escape could have included playing mind games with the guards or the captor. Manipulation of people is a classic "devious" trait. Devious people do not "break up" with their boyfriends--they drive them away with undesirable habits, excessive clingy-ness, nagging, or even by hiring another girl to woo the boy away. Rather than divorce a wife and have to split the estate, why not slowly drive her to insanity? Or have her kidnapped? Or so forth. It doesn't have to be sadistic. Good people can be devious, too.

The great thing about deviousness is that it is more colorful and unpredictable that simple intelligence. It also lends itself to humor. And, dumb characters can be devious, too, in their own way (even unintentionally), so this can level a playing field.

Telepathy:

Examples: Professor Xavier and Jean Grey from X-Men, Matt Parkman from Heroes, Jedi/Sith

Description: This talent is gives the character the ability to read minds and sometimes plant thoughts. It has little to do with actual intelligence, but it gives the character the ability to know anything any other character knows. This can give the illusion of all the above skills combined.

How to Write One: This talent usually gets relegated to super heroes, magical worlds, supernatural worlds, etc. However, it is something that has been extensively researched by assorted militaries, and the idea of telepathy is filtering into popular "realistic" science fiction as well (StarCraft, Firefly). Be aware of your genre and what your readers will or won't reject.

Since telepaths can read minds, they can pull any facts, observations, deductions, ideas, etc. from other characters. This makes them functionally almost as smart as anyone else. Telepaths see things coming and prepare accordingly. They can read emotions, so they are exceptional at mind games and manipulation whether or not they can do "Jedi mind tricks".

There are a lot of things to consider when writing a telepath, like how to not make them completely over-powered. A person who can read and "write" minds will need "Kryptonite". They need a weakness, or their opponents need some sort of shield or counter attack.

Also, what is the extent of their skills? Consider the following:

-Gut instinct (For the purpose of writing, this is like the bottom of the barrel telepathy. This is the cop who can tell what people are trustworthy, the detective who knows when people are lying, the soldier who gets a bad feeling before an ambush.)

-Empathy (This would be a person who can tell when someone is nervous or happy, even if they can't see the person. They can feel what the other person feels.)

-Intuition/Low-Level Mind Reading (Here we have the slight ability to predict others. You see this in people who finish each other's sentences. You can say, "Hey, you remember the thing on the thing that one time we went to that thing?" And they know exactly what you mean. Sometimes.)

-Medium-Level Mind Reading (This would be the person who can read exact thoughts at whim. If you're telling them about a cat and you're thinking about a dog, they would be able to tell you what breed of dog, what color, etc.)

-High-Level Mind Reading (This is the person who can dig into not just your thoughts, but your memories. Stuff you aren't even thinking of. No secret is safe!)

-Manipulator (The Jedi mind trick. This person can influence the behavior of the weak-minded. If they are powerful, maybe even the strong-minded.)

-Implanter (By far the scariest character, this person can erase and replace your memories, leaving you with no way to be certain that you know what you're doing, why you are doing it, or even who you are.)

Additional limitations:

-Range (Do they have to be touching their "victim"? With ten feet? Anywhere in the world? Line of sight?)

-Taxation (Does their ability drain them mentally or physically? What do they have to trade for their ability? Anything?)

-Mental fortitude (Can others with strong wills fend them off?)

-Physical interference (Does a tin foil hat ward them off? How about Magneto's helmet?)

-Physical tools (Is their power bestowed by a machine or magic item?)

-Nemesis (Is there a particular person they cannot affect, or particular people who have the ability to stop them?)

I hope this reference is helpful. If you have any questions, or think something should be added/revised on here, let me know.