Genos stalking Saitama for his training secrets while a grudge-holder circles both of them creates the show's entire comedy in miniature: everyone's obsessed with cracking Saitama open, and he's just trying to live. These specials let ONE write character moments the main series doesn't have time for, especially Genos's relentless desperation and the side heroes actually being funny instead of background decoration. Short enough to rewatch, focused enough that every bit lands.
One-Punch Man received six extra episodes, packaged with its home video volumes, and each one is a fresh story from the manga's author, ONE. The first is set after Saitama battles Mosquito Girl. Genos keeps shadowing Saitama in a suspicious way, determined to learn what makes him so strong. A mysterious stranger also appears, harboring some specific grudge against Saitama. Titles of the later episodes hint at a disciple, a ninja, Bang, a pair of siblings and a murder case. Expect action comedy with sci-fi and supernatural touches.