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Genres

Anime and manga genres differ not only from Western genres in general, but the Japanese concepts of the genres can differ from the English-language fandom's. Also, much more research needs to be done in this area. What follows are a few anime and manga genres and themes which are nonexistent or different in Western tradition.

General

Fantasy
As a genre rather than a mode, series set in complete otherworlds that generally adopt conventions from tabletop, computer, and console RPGs.
Magical Girl
Girls who transform into a magical, superhero-like form. Series generally feature a girl protecting an unaware populace from the monster of the week and, eventually, their leader/general/summoner. A subgenre of henshin ("transformation").
Mecha
Giant robots. Some--the subgenre containing "real robots"--are fairly hard Science Fiction; others--much of the subgenre containing "super robots"--are closer to technobabble-clothed fantasy.
Monster/Summon Battle
Characters fighting via magical or monstrous proxies. Proxies range from pokemon to intelligent, magical dolls in gothic lolita outfits. At least one series arguably uses human proxies. (A subgenre of fighting series.)
Shoujo
Works for girls; notable identifiers include incredibly complex networks of romantic relationships. Any combat is far more likely to be magical than in shounen, and almost never directly against another person.
Shoujo-ai/Yuri
As shounen-ai/yaoi, but substitute "girls" for "boys." Often but not always aimed at female readers. (Exceptions tend to be secondary plots in series designed to titillate male readers.) Not as popular in the U.S. as shounen-ai.
Shounen
Works for boys; notable identifiers include characters who desire to win or be the best at something (passion and force of will). Fighting and sports are common genres, especially together--shounen fighting series tend to include a tournament story arc.
Shounen-ai/Yaoi
(In Japanese, these genres are a single one called "Boys Love" or "BL".) Male homosexual relationships ("shounen-ai") and sex ("yaoi") in stories aimed at female readers. A subgenre of romance. Popular with high-school girls.
Space Battleship
Series set among the crew of a battleship or similar miniworld in outer space. Generally character-driven, although some sort of armed conflict (and questions arising from it) are also standard.
Supernatural
Generally speaking, fantasy intruding on reality and characters dealing with such. (e.g., ghostbusting)

Specific Creators

Some specific creators are functionally their own genres. A few important such are given below.

CLAMP
An all-female manga studio. Their works are frequently adapted into anime. Highly stylized art, references to other works of theirs, and tragedy pervade much of their oeuvre, which is almost entirely fantasy.
Leiji Matsumoto
A highly unusual but well-loved anime and manga creator, primarily of science fiction with fantasy influences, with a very distinct visual style and a sprawling continuity (of sorts). Responsible for a number of iconic characters, starships, and logos.
Hayao Miyazaki
Children's movie writer/director/animator/producer, almost solely of fantasy films. His films manage the rare feat of both winning critical acclaim (and major film awards) and breaking box-office records--one film was the highest-grossing movie of all time in Japan until Titanic came out. His films generally advocate feminism, environmentalism, and pacifism. Flight is also a strong theme. Strongly influenced by Western fantasy. Miyazaki recently retired from feature films for the second time.
Osamu Tezuka
Known as "the father of Japanese animation" and "The God of Manga." Much of his work isn't fantasy, but some of it is. Often wrote for children. (With a total output of over 700 works by the time of his death at age 60, no accurate overarching generalizations can be made.) Some of his works are accused of including racist caricatures in the character designs, but opinion varies.