Even though he was a middle-aged man of 46 when the show began, Lou Costello did most of his own stunts on the show. An athlete in his youth, he was actually a stuntman in Hollywood for a time back in the silent era before he teamed up with partner Bud Abbott, and was renowned for taking spectacular pratfalls in his films and on stage. Stuntmen were used for the more potentially dangerous stunts--being knocked through walls, getting hit by cars, etc.--but most of the falls you see Costello take were actually done by him. For example, in the episode "The Tax Return", there's a scene in which two crooks break into Bud & Lou's apartment, and a rather knock-down, drag-out brawl erupts. Although it looks like a stuntman is doubling for Lou in the fight scene, at one point the "stuntman" turns around and it is very clear that it actually is Costello doing the fighting.
In the first season, Bud Abbott and Lou Costello had a pet chimp named Bingo the Chimp. Costello didn't particularly like Bingo and apparently Bingo sensed it, because while they were filming a scene one day, Bingo turned and bit him. Costello demanded that Bingo be fired, and since his company was producing the show, Bingo was gone the next week and was never mentioned in the series again.
During filming, one camera was always kept on Lou Costello because he was constantly improvising. The funniest bits of business were then edited into the episode whether they had anything to do with the storyline or not.