The name of Groucho Marx's character, "S. Quentin Quale", caused a stir when the film was first released due to the subtle but clear joke: the use of the term "San Quentin quail", which means "jail bait".
Winston Churchill was informed of Rudolf Hess' capture in Scotland just before he was about to see a private screening of this film. Spurning a more detailed briefing, Churchill famously commented, "Well, Hess or no Hess, I'm off to see The Marx Brothers". (Different versions of this story insist the film Churchill was to see was Monkey Business (1931) or The Big Store (1941).)
The elaborate chase climax was going to be deleted before filming because MGM execs thought it was too expensive.
When Chico suggests they use a telephone, Groucho says "This is 1870; Don Ameche has not invented the telephone yet." This joke refers to the film "The Story of Alexander Graham Bell", starring Don Ameche, released one year prior to this film.