Trivia: The entire film was filmed in sequence. Meaning that each shot you see in the movie is the order in which the filmmakers filmed it. This is unusual for films as filming in sequence can be very inefficient, as the filmmakers have to keep revisiting the same sets and setting them up more than once rather than just filming all their scenes while they have the set ready for it.
Trivia: When Saul comes to visit John, John plays a joke on him by pointing to an empty chair and saying, "You've met Harvey." This is a reference to a play by Mary Chase titled "Harvey" in which Harvey is an imaginary rabbit seen by the main character.
Trivia: According to a 2001 Entertainment Weekly (American entertainment magazine) article on this film, the filmmakers originally wanted to mention Nash's homosexuality, but they feared the film will make the wrong connection between homosexuality and schizophrenia, so they abandoned it. This connection, according to the article, was based on several now-discredited psychological studies that first appeared in the late 1950s.