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Dating sims (or dating simulations) are a video game subgenre of simulation games, usually Japanese, with romantic elements. The most common objective of dating sims is to date, usually choosing from among several characters, and to achieve a romantic relationship.

An otome game is a story based video game that is targeted towards the female market. Generally one of the goals, besides the main plot goal, is to develop a romantic relationship between the female player character and one of several male or female characters.

http://en.wikipedia.org/wiki/Otome_game

Yaoi, also known as boys' love, is a genre of work involving romance between men. The word originates in Japan and is typically seen alongside an anime style.