Gene Kelly discovered Leslie Caron dancing ballet while he was on vacation in Paris. Caron spoke very little English when she made the movie. Fortunately, as IMDb puts it, she was fluent in dance. (Source)

If you were alive in 1988 and had a spare $15,000 lying around, you could've bought the Oscar that An American in Paris won for Best Picture. That's how much it sold for at auction. (Source)

Nina Fochs (Milo) came down with chicken pox during the filming of the movie. When she came back to work, makeup artists had their work cut out for them, as they struggled to cover her pockmarks. (Source)

The dance sequence cost about $450,000 and took one month to make in 1950. That's almost five million bucks in today's money. Still one month, though. (Source)