An alien race known as the Oni arrive on Earth to invade 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 taking part in a competition. The competition is a variant of the game of tag (known as "the game of the Oni" in Japanese), in which the human player must touch the horns on the head of the Oni player within one week. The computer-selected human player is you, a lecherous, unlucky and academically unsuccessful high school student from the fictional Tomobiki Town in Nerima, Japan, and the Oni player is Lum, daughter of the leader of the alien invaders. Despite your initial reluctance to take part in the competition, you becomes interested in the game when you meets Lum. When the competition begins, Lum surprises everyone by flying away and you find yourself unable to catch her. Before the last day of the competition, your girlfriend Shinobu Miyake encourages you by pledging to marry you if you wins. On the final day of the competition, you won the game by stealing Lum's bikini top, which prevents her from protecting her horns in favor of protecting her modesty. In celebrating your victory, you expresses your joy at being able to get married; however, Lum misinterprets this as a proposal from you and accepts on the competition's worldwide live television broadcast. Despite the misunderstanding, Lum falls in love with you and moves into your house.

Despite your lack of interest in Lum and attempts to rekindle your relationship with Shinobu, Lum frequently interferes. Later Lum begins attending the same school as you despite your objections.