The title is derived from Miguel de Cervantes' novel "Don Quixote." Quixote tilted at windmills because he was sure they were giants. In this movie, Justin states that Quixote was crazy to be convinced that windmills are giants. But to wonder if "they might be giants" - that's another matter.
In later years, Joanne Woodward claimed that she had had such an unhappy time making this film that she had actually contemplated giving up her acting career. She would not be drawn further on the topic, beyond saying that her unhappiness had nothing to do with her co-star George C. Scott, whom she described as "a gentleman".
The genesis of this film is unusual, as James Goldman's original play, dating from the early-60s, had never been produced anywhere in the United States by the time the film was made. Goldman had reworked it several times, and there had been a short-run production of it in London in 1961, directed by Joan Littlewood at her famous Stratford East theater, far away from the West End. Harry H. Corbett had played the lead. Goldman was still dissatisfied, but, after the great success in 1968 of the film version of his subsequent play "The Lion In Winter", he and its director Anthony Harvey found themselves being lauded in Hollywood and asked if they might have any ideas for a future collaboration. Goldman had another go at revising his play as a film script, now setting it in New York streets and creating many new characters. The film was not a success, although it has become a cult favorite.
This film provided US Alternative rock outfit 'They Might be Giants' with their name.