An alien race known as the Oni arrive on Earth with the intention of invading the planet. Instead of taking over the planet by force, the Oni give humans a chance to fight for the rights to the planet by competing in a competition. The competition is a variant of the game of tag, in which the human player must touch the horns on the head of the Oni player within one week. The chosen human player is Ataru Moroboshi, a lecherous high school student, and the Oni player is Lum, daughter of the leader of the attempted Invasion.
Despite his initial reluctance to take part in the competition, Ataru becomes extremely interested in the game when he meets Lum. When the competition begins, Lum surprises everyone by flying away and Ataru finds himself unable to catch her. Before the last day of the competition, Ataru's girlfriend Shinobu Miyake encourages Ataru by pledging to marry him if he wins. On the final day of the competition, Ataru wins the game by stealing Lum's bikini top, which prevents her from protecting her horns in favour of protecting her modesty. In celebrating his victory, Ataru expresses his joy at being able to get married; however, Lum misinterprets this as a proposal from Ataru and accepts on live television. Despite the misunderstanding, Lum falls in love with Ataru and moves into his house.
Despite Ataru's lack of interest in Lum and attempts to rekindle his relationship with Shinobu, Lum frequently interferes and eventually Shinobu loses interest in Ataru. Still, Ataru's flirtatious nature persists despite Lum's constant attention. Lum attempts to stop him flirting, which results in Ataru receiving electric shock attacks from Lum as punishment. Lum begins attending the same school as Ataru despite his objections, and Lum develops a fanbase of admirers among the boys of the school, including Shutaro Mendou, the heir to a large corporation. Despite their romantic interest, none of Lum's admirers would risk upsetting Lum by trying to force her and Ataru apart.