http://tvtropes.org/pmwiki/pmwiki.php/Main/Fanon

Actor Sokka, Avatar: The Last Airbender. "Oh, no — another fan with ideas..."

Official canon is much smaller than the people who throw the term around like to think it is. Canonicity is limited to that which has actually been described in the source material, though if one must stretch the definition, it may also include anything the creators only mention in interviews or supplementary material. But ultimately, canon boils down to what the creators specifically need to worry about conveying to the audience for the purposes of the ongoing plot. In short, canon is what the work clearly says about itself.

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Fanon, also known by the term "headcanon"note though this term refers solely to personal views on the work, rather than wider-held fan theories and beliefs, is the set of theories based on that source material which, no matter how much they seem to be the "obvious" or "only" interpretation of canonical fact, are not actually part of the canon. It can arise in a few ways, but mostly boils down to clarifying any vagueness in a story and/or trying to justify an idea held about an element of a work. A piece of fanon can come about to explain holes in the story that the creators of a work may have either deliberately left open in order to have fodder for later stories, or never gave any thought to because it was considered an unimportant side detail. Or it can come into existence as a fact gained from a popular but non-canonical source or a different adaptation of the work.

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Because many fans mistake their own fanon for actual canon, they tend to get riled up when a new fact is introduced which does not literally contradict anything canonical, but invalidates what were formerly the most obvious assumptions. Many examples of Retcon and Continuity Drift that are imagined to be violations of canon really only explicitly contradict fanon. In this way, when someone says "That episode was terrible because it violates the canon!", there is a good chance that it only violates their personal canon.

Popular subjects of fanon include character backstories, full names of characters with No Name Given, what characters actually do for a living, and Shipping  a whole other world of its own.

Since many creators in the aftermarket series universe are fans, fanon often shows up there, and if those creators in turn start writing for the main continuity, fanon may actually become canonical. Alternatively, you just have Memetic Mutation within the fandom.

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Remember up above, where we mentioned stretching the definition of canon? Well, if you don't stretch it, fanon can also refer to the body of information provided by otherwise-official sources outside of the main work's continuity. Television and movie scripts are a continuing source of fanon material  Captain James Kirk, for example, has the middle initial ("T.")... but his actual middle name of "Tiberius" was originally revealed in an episode of the Trek animated series; since that show's canonicity is debatable, it was considered "fanon" until featured and explained in the novelisation of the first Star Trek movie, which was written by creator Gene Roddenberry and therefore canonical. It was also stated explicitly in the sixth movie. Note that this usage blurs the line between fanon and deuterocanon, though.

Warning: Fanon and accusations of fanon are a classic Flame Bait, with the accusation commonly leveled by fans who have a different interpretation of the material — even when their theory is just as vulnerable to Schrödinger's Gun.

Compare Broad Strokes, where the events of a story are referenced in passing without taking everything said and done as having "officially" happened. If the fanon was repeatedly hinted at by writers until it became fanon, but never actually confirmed in canon, it's Writer-Induced Fanon. See also Fandom-Specific Plot and Recurring Fanon Character, when a fan-created Original Character becomes sufficiently popular as to be widely used by other fan creators or even mistaken for canon. Not to be confused with this Fanon or the Pope's robe. Fanon is frequently based on Fanfic Fuel.

Examples:

Franchise-Specific Fanons:

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Music

Vocaloid: The only things officially canonical are the characters' names, appearances, and voices. Usually. (Some have a couple more minor things, like age and height, while others only have a name and a voice.) Personalities, backstories, relationships, and some characters are pure fanon.

Pink Floyd: Fans have generally accepted that Pink, the protagonist of The Wall, was born "Floyd Pinkerton", and that "Pink Floyd" is his stage name. In the original album, he's only referred to as "Pink" and "Mr. Floyd" in two separate moments, but the movie adaptation includes a brief scene where his deceased father's name is given on a memorial plaque as "J.H. Pinkerton", and one of Pink's friends can be heard calling him "Pinky" (a logical nickname for someone with the surname "Pinkerton") in another scene. Though somewhat less unanimous than the above theory, many fans also believe that The Final Cut, Roger Waters' final album with the band, is partially an epilogue/continuation of The Wall. In particular, many have theorized that the titular song, "The Final Cut", is told from Pink's perspective as he contemplates suicide sometime after recovering from his mental breakdown, and that "When the Tigers Broke Free" is about the death of his father. The latter point is supported by the film version of The Wall, which actually includes "When the Tigers Broke" in the soundtrack, but it's unknown if Waters actually wrote the song with Pink in mind.

"Fuck You" by Archive is about someones utter disdain for another. Due to a line late in the song most believe it's referring to the singer himself, warping it into a song about severe self-hatred.

The female voice in "Sad Machine" by Porter Robinson is often assumed to be Rin, the protagonist in "Shelter" because of both of them existing in a virtual space who end up connecting to the outside world over the course of the songs.

Nobody seems to know where the theory that R.E.M.'s "Shiny Happy People" is about Chinese communist propaganda comes from, but it certainly doesn't come from the band themselves. They have often openly regretted writing the song but they've never tried to justify it by ascribing any deeper meaning to it. It hasn't stopped other people from trying to do so.

Dion DiMucci's song "The Wanderer" is a Sequel Song to "Runaround Sue".

Rachel Rose Mitchell's "Hey Alice" is Villain Song from the perspective of someone patrionizing Alice of Alice In Wonderland. Fans theorize that Alice is being sent to a Bedlam Asylum and the singer is either her sister or her therapist. Another interpretation is that it's about Alice choosing between sanity and madness (or fantasy and reality).

Newspaper Comics

Many fans believe that Garfield and Jim Davis's lesser-known second strip, U.S. Acres, take place in the same universe. While this is true on the Garfield and Friends side, it's never been confirmed or debunked in the strips.

Tabletop Games

Theater

Hamlet is a breeding ground for these, due partly to centuries of theatrical interpretation and partly to Everyone Is Jesus in Purgatory induced by high school English classes. Thus, for instance, many people take it for granted that Hamlet is genuinely mad instead of faking it.

The Little Shop of Horrors fandom often gives Audrey's full name as Audrey Fulquard. This surname is from The Little Shop of Horrors, the B-movie on which the musical is—very loosely—based.

Theme Parks

Disney's Haunted Mansion hasn't got any real official backstory so far, although there are just enough clues to make the audience wonder. The Internet has a gigantic amount of theories about the backstory of the Mansion. More recent additions to the Mansion seem to create a "canonical" backstory have raised great protestation from the fans who imagined a backstory as they rode through the attraction.

Visual Novels

Nasuverse: "The Moonlit World" isn't what the Magical Society is called in official works. It originated as the name of the website that formerly hosted the Nasuverse fan forum Beast's Lair. When the site was rebooted after an untimely crash, the website portion of the forum was lost. The term "Moonlit World", however, was such a unique way to describe The Masquerade and its many wonders and dangers that fans kept it around for their fanworks and forum discussions.

Umineko: When They Cry is a case where Fanon is actually encouraged by the author. Ryūkishi wants the readers to have their own ideas and reach their own conclusions about the plot and the characters. Very few answers about the mysteries are revealed explicitly; but most readers end up acknowledging a certain conclusion (namely that Beatrice, Kanon and Shannon are one and the same ) even if the story never outright states it. The manga tends to be a bit more explicit on some aspects though.

Web Original