Kid Fic

Fanfic genre based on the idea of giving one or more characters a child to raise. This is a common device for promoting a favored Shipping (often slash), by showing how well the favored pair act as parents together. Sometimes the resident Badass will be given a child to soften them (can be seen as somewhat related to Morality Pet) before pairing them with The Hero.

In some cases the child may be another canon character, de-aged in some manner. In other cases, it can be the instantiation of an Intercontinuity Crossover, with a child imported from another story or series.

When one or more males are involved in the process, mpreg is always a possibility.

If the author is particularly unoriginal, Patchwork Kids or Gender Equals Breed may ensue.

If this is canonical, it may be a Parents for a Day situation.

Add a few years, and you get Next Gen Fic.

Anime & Manga

 * The Lucky Ones, an Inuyasha fanfiction by Terri Bota, is one of the most well-known.
 * In WAFF Death Note fics L and Light often end up raising Mello, Matt, and Near.
 * Here's a Death Note Kid Fic that is closer to the tone of the original story.
 * Anime in general tend to be susceptible to 'chibification', where one or more characters are somehow turned into chibis and/or children and must be attended to by another character before the chibification wears off.

Literature

 * Laume (on fanfiction.net) has written several kidfics for Harry Potter, including:
 * Last Chance, in which a slightly more sensitive Dumbledore ends up raising the young Tom Riddle, and
 * Four and Counting, in which a potions mishap reduces Severus Snape to age four, several years after Voldemort's defeat.
 * A very well-written example would be Stealing Harry by Sam The Story Teller (link), which starts with Harry being raised by first the Dursleys and then Remus and Sirius (who open a bookshop near the Dursleys' house to keep a watch on him, then kidnap him after discovering he was being abused).
 * This Harry Potter fanfiction, called Baby Project, is a perfect example.
 * Also this classic Dramione fic, Future Parents Program.
 * Malfoy Child, is a classic Drarry fic in which Draco is de-aged and put into Harry's care. It is actually really good.

Live Action TV

 * In the Leverage fandom on livejournal there's a somewhat common trend of de-aging the team's badass bruiser Eliot to be a small child and then sticking him under Nate's care.
 * Considering the Badass Spencer is? If someone competent works with it, it sounds like it could be Crowning Moment of Funny. Bonus points if they manage to de-age Asberger's case Parker along with...
 * Blessed Is His Child, a Supernatural kidfic, of all things that manages to be hilarious and heartbreaking within the space of a chapter! (pairings Dean/Cas, Sam/Gabriel and Crowley/Anna, as well as various OC pairings)
 * Links. Now. Needed.
 * Here's part one, which links to the rest. Be warned, it's unfinished and may well be abandoned.
 * And here's the prologue. If you read it, you'll be on the same page as the angels; if you don't, you'll be on the same page as the Winchesters.
 * Another Supernatural Kidfic series: The Adventures of John Gabriel Winchester, The Messiah. about Dean and Castiel's son. The pairings are Dean/Cas and Sam/Ruby.
 * There are a LOT of House finding out he has a daughter to the point of the fans getting annoyed at them.
 * The Lizzyverse, a Sherlock series, is about Sherlock and John's adopted daughter Elizabeth. It starts out with how they go about scaring off her boyfriends, and skips all over the age range after that.
 * Getting Better is an AU in which both Sherlock and John are single parents, centred around the growing friendship between their children, Emily and Tristram.
 * In this drabble, Sherlock's abused son shows up on their doorstep. No pairings.
 * The Nursery Files is an entire website dedicated to The X-Files kidfics from various couples in various categories.
 * A few writers in the iCarly fandom write nothing but this.
 * White Collar gets two with a de-aged Neal for Peter and Elizabeth to raise: One with him as a teenager and another with him as a toddler.
 * There is a whole website dedicated to Stargate SG-1 and Stargate Atlantis kidfic, mostly of the deaged!character type: The Littleverse.

Web Comics

 * There are a few Homestuck fanfics based on this, such as "Cold Grublings". Made very convenient by the setting's local Applied Phlebotinum - "ectobiology", which allows an instant creation of a child which is a genetic mix of any two persons from anywhere and anytime (regardless of gender or age), without the need for actual physical contact.
 * The universe introduced in Act 6 created a lot of these, as it had

Film

 * So Wise We Grow is a Kirk/Spock Star Trek story in which the Vulcan Council locates Spock's son among the surviving Vulcans. Which would be great, except Spock's never heard of the kid before in his life.
 * Another Star Trek is the Trust series; in which prior to his marriage to Spock, Sufficiently Advanced Aliens force Kirk to address his deep-seated trust issues by regressing into a troubled childhood. It's up to Spock and Kirk's elder brother Sam to help him get back to normal. Handles the subject matter more tastefully than one would expect, and includes social commentary on the TOS-canon Federation's system of term-marriages.
 * Fanfiction.net has a Star Trek community dedicated to archiving de-aged fics, fic about the crew's childhoods, and childhood A Us. Mostly de-aged fic.

Comics

 * All That Remains in which Tony Stark takes in the female clone of Captain America (comics) after the Marvel Civil War.
 * Ice Cream Ice Cream by Ash Ai is a series of one shots that has Tony and Pepper raising their their little kid, Jared.

Western Animation

 * Thundercracker's Glory by Kenya Starflight is a Transformers Generation 1 fic in which Thundercracker finds himself caretaker to his orphaned niece after his co-creation (sister) is killed. Shenanigans ensue. A sequel is possibly in the works...

Other

 * Kijikun (on livejournal) has written several kidfics for Biker Mice From Mars, The Incredibles, Highlander, Due South, and Iron Man/Captain America (comics).
 * The Redemption Clause which follows the growing relationship between Buddy Pine and a adult Violet Parr, partly centered on Buddy Pine's adopted daughter.
 * A good example of a crossover Kid Fic would be Clell65619's Harry Potter and the Invincible Technomage, in which Tony Stark (and by extension, many of the premiere superheroes of the Marvel Universe) raises a young Harry Potter.
 * The Son of Mine series on Fan Fiction.Net takes places in the Guyver universe, and gives Masaki Murakami a son.
 * Common in Touhou doujins. Some of the more notable ones include Life of Maid and Mamange (Danbooru warning) as well as quite a few others.