Links: TVTropes, Wikipedia, IMDB (8.6).
I'm probably going to take some flak for this, but... I actively disliked this movie. This Martin Scorsese film is set in the grimiest portrayal of New York City you're likely to encounter, and the film is two hours of focused character study directed at Robert De Niro's title character: an unlikable, socially awkward protagonist who's visibly descending into homicidal, stalkeriffic madness. The film is very unpleasant to watch, especially the stalking scenes which hewed so sharply to reality that they nicked it, but I felt there was no good payoff for that unpleasantness. In fact, the end makes more sense as a fantasy or dream ending than it does at face value. It doesn't help that, while it isn't quite Film Noir, it draws enough of its inspiration from that well of lingering, jazz-infested night shots that my dislike of Film Noir as a style kicks in. Score: 1 star out of 4, unless you like Film Noir and dark character studies (in which case this is a classic).
Aside: if you're not aware, this is the movie that inspired John Hinckley Jr. to shoot Ronald Reagan in 1981, which he believed would impress Jodie Foster (who plays a child prostitute in this film). Part of what frustrates me about the film is that it presents the crazy without judging it; would Hinckley have been so inspired if the movie had ended on a sour note instead?