Mutants

Beware the mutant.

Mutants are, by general definition, organisms that have undergone a permanent change to their genetic structure relative to the norm for their species. Sometimes this results in a new race or breed, or even, as mutations accumulate over time, a new species (speciation), and sometimes it's a one-off that produces effects that don't breed true, or are (most commonly) so negative that they prevent the individual mutant from successfully surviving and breeding. Technically, any deviation in a person's genetic code from a "simple" combination of his parents' alleles would be a mutation. On average, human beings have 150 to 175 mutations each, the vast majority of which are undetectable. So technically, we're all mutants.

That's in real life.

Mutants in fiction follow this same basic idea (though normally portrayed in biologically implausible or impossible ways), but have a much wider variety of phenotype effects than Real Life mutations because of Hollywood Evolution. Fictional mutants are often Super Heroes, because "mutation" is a very easy way to get multiple power sets from a single Super-Hero Origin (this is why X-Men, the first major comic to have Loads and Loads of Characters introduced all at once, starred a group of mutants). Just as often, humans or other animals are mutated somehow into hideous Exclusively Evil monsters for the heroes to fight, or pitiful misunderstood freaks.

There are two different ways to use the word Mutant: it can refer to characters who were born with a mutation, such as most members of the X-Men. Marvel Comics, the storytellers that have to be the most specific about this kind of thing, refers to these as Mutants. The other definition is things that have had their genetic code changed after birth, whether by radiation damage, gene therapy, or a pandemic disease. Marvel calls these Mutates. Usually in fiction, these after-market mutations also alter germ cells and get passed on to the mutate's kids. For clarity's sake (and in accordance with the Mutant Registration Act), we keep the examples of these two different kinds of mutation separate.

May overlap with Evolutionary Levels if the mutants are "superior" to mankind. A type of Super-Hero Origin.

Examples of Mutants include:

True Mutants

Anime

The diclonii in Elfen Lied. A particularly creepy example, given that after going through the motions of persecution and acceptance, humanity exterminates them and this is considered a good thing.
The idea that Newtypes in Mobile Suit Gundam are a result of evolution or mutation is gradually debunked as the metaseries goes on. In the earliest shows, however, they are implied to be humans adapting to living (and dying) in space by gaining enhanced senses and low-level Psychic Powers. All related tropes about mutants being exploited, persecuted, and persecuting back apply in full, as well.
Sein of Magical Girl Lyrical Nanoha was revealed in the third Sound Stage of StrikerS to be one in addition to being a Cyborg. Unlike her sisters, she had an unexpected mutation when she was being born. This is the source of her phasing abilities.
The Contractors and Dolls of Darker than Black probably count, being a Darker and Edgier take on the traditional comic books version, although other than the fact that they generally have a Meta Origin, there's no indication of whether or not any genetic difference exists.
- It might even be a subversion; it's mentioned offhand that some government research organizations assumed it was genetic, tried eugenics programs to get themselves a superpowered army, and failed miserably.

Comic Books

This is the premise of the X-Men franchise. where mutations form the team's Meta Origin for Stock Super Powers ranging from the semi plausible to the blatantly absurd. Discrimination against Mutants and by Mutants (some of whom believe that they, collectively, are a step above humanity) is one of the work's constant major themes.
The Dark Knight comic had a massive teen gang called the Mutants terrifying Gotham. The leader was probably the only Mutant who might actually be a mutant. Well, that or a plain old sociopath with filed teeth.
An issue of the classic EC Comics horror/sci-fi anthology series Weird Science (not that one) had a story titled "The Loathsome" about a badly deformed little girl being raised in an orphanage. Her father, a US Navy sailor, had his own DNA damaged by exposure to radiation during an atom bomb test in the Pacific just prior to returning home and fathering her. A military doctor convinced him to give up the infant and tell his wife the child had died shortly after birth. The girl had no super powers and certainly wasn't evil, but was treated as a monster by her caretakers and the other orphans because of her severe physical deformities. The story was An Aesop (a heart-wrenchingly tragic one) about the perils of nuclear weapons and mistreatment of people who are "different." You can read it here.
- Paid homage to by Alex Ross with Maggie in Marvels (See above image.)

Film

The film Freaks which also contains a Real Life example since no makeup or special effects were used. The stars of this movie were sadly plagued with rare disabilities but were apparently happy and willing to travel with the circus and make a career out of it.
The giant ants in Them! are an example, oddly enough. The radiation from nuclear testing didn't cause the ants to suddenly become gigantic; it was the offspring of those irradiated ants (along with their increased size, they emerged as adults from their eggs, with no larval or pupal stages in between. Another genetic quirk).
The telepathic mutants in Beneath The Planet Of The Apes.

Literature

Giant Exclusively Evil Mutants are a staple of older pulp science fiction, serving the same antagonistic purpose as goblins or trolls do in fantasy.
Perry Rhodan used to feature a Mutant Corps (with generally Psychic Powers) especially in the early series, when they were one of Earth's few trump cards against a militarily superior universe. Over time, attrition took its toll (especially when the first-generation immortality phlebotinum became unavailable and the replacement was sharply limited in supply) and various countermeasures were introduced, but the few remaining mutants stayed on as main characters for a long time and occasionally a new one or two would show up. More recently, a wave of mostly teenage mutants appeared that could trace their origin back to mass genetic engineering on a particular colony world during the latest dark age of the galaxy; these were, however, also afflicted with a deliberate genetic flaw that would eventually trigger and kill them, and the eventual cure had the side effect of removing their powers as well.
The Foundation series gives us The Mule, although the later novels retconned his origin.
The escapism inherent in this trope was subverted as early as 1954, in Alfred Bester's short story "5,271,009." Here, the main character is put in a Lotus Eater Machine and experiences multiple juvenile fantasies, each of which is explained by "a mysterious mutant strain in his makeup that makes him different." On the other hand, the Espers (telepaths) of The Demolished Man fit the X-Men version very closely, including the idea of classifying them according to different levels of super power.
The Vorkosigan Saga actually uses mutants correctly: instead of superpowers, you get horrible deformities! Due to intense radiation on Barrayar, children born with grotesque mutations were common, and mutants were usually killed at birth. This left a long-standing cultural stigma against all apparent genetic abnormalities, including intentional genetic engineering like the Quaddies and hermaphrodites as well as non-genetic birth defects like a cleft lip.
- Mutations on Barrayar came about because the initial terraforming group was trapped on the planet when the only wormhole leading to the system closed, and the population was not large enough to have sufficient genetic diversity (especially after the initial die-off), so they had major inbreeding problems, which caused continual genetic problems.
Serroi in the Duel of Sorcery Trilogy. In this case, it means that she's tiny, green-skinned, and has what seems to be a pineal eye of sorts and Psychic Powers.
Mutants born after a suitcase bomb goes off in New Jersey are the protagonists (and antagonists) of Tom De Haven's affecting novel Freaks' Amour.
The Chrysalids by John Windham features a society (implied to be post-nuclear apocalypse) where mutants are a common occurrence. However, seeing as its a Crapsack World, they're immediately exiled or killed on discovery.

Live Action TV

The Daleks of Doctor Who. And apparently the Thals, although they seem to have recovered by the start of The Dead Planet. Alydon [speaking of the Daleks] : "If they call us mutations... what must they be like?"
- The Thals are mutants, too. It's just that their mutations were beneficial, making them healthy and beautiful.
  - The Thals are an interesting subversion. They initially mutated into creatures similar to the Daleks, but where the Daleks "stabilized" and stopped mutating the Thals continued, eventually coming full circle, mutating back into basically their original form.
Heroes presents its Stock Super Powers as the product of mutation.
The Thematic Rogues Gallery of Power Rangers Time Force.
The X-Files regularly featured these as the Monster of the Week, including the liver-eating Eugene Victor Tooms, among numerous others.
Sanctuary has Abnormals, which seems to cover mutants and mutates.
Played for comedy and mixed with Lego Genetics in one episode of The Mighty Boosh. We are the mutant race! Don't look at my eyes, don't look at my face!
Gwen Raiden from Angel has extremely high voltage, and over the years has discovered interesting uses with her powers. For example, sticking her hand in a laser grid, and taking control of the lasers without setting off the alarm. She stands out as this trope because she lacks demonic heritage or magical training like most powered being in this universe, she just developed it naturally. (Though the source of her power is never really examined and could in fact be supernaturally based. All that's really known is she gets struck by lightning a lot.)
- There was also a character named Bethany who had telekinesis, but that might be considered a Psychic Power instead. Again, precisely where the power comes from is not explained, but it's only present in people who suffer severe trauma.
The main focus of Mutant X

Tabletop Games

In Paranoia, Mutants tie with Commies and Traitors for the greatest threat to Alpha Complex. In fact, if you are a Mutant, you probably are a Commie Traitor as well. Unfortunately, every Player Character is a mutant. Please report to the nearest Execution Center, citizen.
- Being a mutant is just grounds for immediate execution. The only exception being if you are a 'registered mutant' in which case you just have to wear a bright yellow ribbon at all times, and might as well have "scapegoat" tattooed on your head.
D20 Modern features rules for playing as someone with special mutant abilities.
Gamma World had an After the End setting with mutated humans, animals, and plants.
Mutant and Mutant Space were Swedish roleplaying games that spawned the Mutant Chronicles universe (despite the name, MC is not about mutants).
- However, Mutant has been re-released several time afterwards. It is, after all, Sweden's third or second largest domestic tabletop game. And for any reader who is confused; Mutant is not about mutants, but they sure as h*ll are an important part of it.
Mutants and Masterminds uses this as a common 'origin story'.
Warhammer 40,000's Imperium of Man views mutation as a scourge on the purity of the human race, and a threat to the very survival of the species. While the official stance on mutation calls for harsh measures, many worlds nonetheless have sizable mutant populations, who are either wretched outcasts scavenging an existence on the fringes of society, or used as highly-disposable slave labor or Cannon Fodder. However, some (dozens, though most are obscure, and manyused to live in only one place and died out) strains of mutants have stabilized into subraces the Imperium calls "abhumans," most notably Ogryns and Ratlings, who serve specialized roles in the Imperial Guard, and the psychic Navigator caste who are required for Warp travel.
Mutants in Warhammer Fantasy Battle are usually killed at birth or killed later by the Witch Hunters, but those that do survive tend to retreat to the dark corners of the Old World and band together with similar misshapen outcasts. The beastmen are one such society, a successful "breed" of mutant that resembles a savage, bloodthirsty satyr.
There's mutants in Exalted too. Mutants come in two main flavors, the bioengineered people created by the Solars in the First Age and the humans and creatures exposed to the Wyld at the edge of the world. The first are somewhat rational, physically predictable and still human in their way of thinking, the second... can vary widely in sanity and form.
- Oh, and there are people whose bodies are twisted by power places like demenses or the fact they have inhuman ancestry from gods, demons and other creatures...
Palladium Games' After the Bomb, similar to the Fallout example below, has the Crash so effect human beings that they evolve into mutant animals. The Empire of Humanity places great stock in keeping the gene pool pure, but most of the planet is at least a little mutated.

Video Games

City of Heroes has mutation as one of the stock superpower origins.
The Sai, the descendants of the humans in the soon-to-be-released game Stormrise have mutated in order to adapt to the extreme conditions after a experiment to change the climate of earth by mankind went horribly wrong, resulting in the mother of all storms and turning most the earth uninhabitable.
In Nasuverse a lot of characters get their powers by being born mutants. In one of the side stories it is said that while such a person can turn the side of a battle, they typically don't survive said battle. And keeping these mutations in the next generation is a difficult task. A few families of the Demon Hunter Association managed to overcome these problems, especially Nanaya and Ryogi. Their powers were not particularly impressive, but they compensated this by being Badass Families.
In Killzone the Helghast have evolved to suit the environment of their adopted homeworld, to the point that they require specialized breathing apparatus to visit other worlds. Complete with evil red eyes.
In The Final Fantasy Legend, mutants (also referred to as espers in the remake of the second game) are one of the four (three in the first game) playable races. They randomly gain or lose traits after battles.
X-Com Apocalypse has "Mutant Alliance" organization protecting the rights of... a bunch of big-eyed, physically frail but psionically strong folk. Of course, there's a reason for such specific traits: it's but a transparent euphemism for "Sectoid hybrid", as explained right in the description of said "Mutant Alliance".
In UFO: Aftershock, there are two races, the Cyborgs and the Psionics, who are humans who were born with special abilities due to a genetic mutation caused by the Biomass. The Cyborgs' mutation allowed them to graft robotic implants onto themselves easily, while the Psionics developed Psychic Powers.

Web Comics

Bug deconstructs the idea of gaining superpowers by mutation.
In The Inexplicable Adventures of Bob, a Unicorn mare was exposed to strange pollution, and her offspring grew into Generictown's resident Kaiju, Unigar the Vast Unicorn.
- And the Loch Ness Monster is a 65-million year old mutant newt.

Web Original

Mutants in the X-Men style are the basis of the Whateley Universe. Some tiny fraction of the people who have the 'meta-gene complex' manifest as mutants, typically around age 14. Most of them - good, bad, or indifferent - go to Whateley Academy to learn how to use their powers and not get murdered by lynch mobs of baselines.

Western Animation

Futurama features a society of downtrodden mutants with amusingly altered physical states (a girl with a pig nose and gills, a man with a third arm instead of an ear) forced to live in the sewers of New New York. None of them have superpowers, though (well, not from their mutations, anyway).

Real Life

The preponderance of evidence suggests that Britain's Queen Victoria was a genuine mutant. Many of her descendants inherited the allele for hemophilia from her, yet neither of her parents' bloodlines carry this trait, indicating that a random genetic mutation made her a carrier for this disorder. ...that assumes that all of her official ancestors were her actual ancestors.
See the Film example on this page.
Elizabeth Taylor was a mutant—not for her famous violet eyes, which were in fact an illusion (she had very deep blue eyes that appeared purple with the right lighting and makeup), but because she had two rows of eyelashes on each eyelid.

Mutates

Anime and Manga

The Zoanoids from the Guyver franchise.
No human examples, but in the Acid Tokyo storyline of Tsubasa Reservoir Chronicle, they spend a fair portion of their time fighting hideous, deformed monsters whom the survivors identify as 'mutants.' How having the world destroyed by acid raid resulted in animals mutating into three-headed spitting snakes or carnivorous earthworms the size of semis is never really explained.