1957's "The Enemy Below" is a really excellent film, and I recommend it highly even if you (like me) aren't a big fan of WWII films in general.

Starring Robert Mitchum as the U.S. warship captain and Curd Jurgens as the German U-boat captain (who we quickly learn is a loyal German but disgusted with Hitler and the Nazis), it also features David Hedison and Theodore Bikel. (Sidenote: Footage from the film was later used by the TV series "Voyage to the Bottom of the Sea", which also featured David Hedison!)

The film tells the story of a intense cat-and-mouse chase between the ship and the sub, with both captains gradually gaining more and more respect for each other, though they of course had never met.

As you watch the film, if the plot starts to seem oddly familiar, it's because the original "Star Trek" episode "Balance of Terror" (1966) was inspired by and in key respects tightly parallels the film.

Fascinating.

@lauren interestingly, the book it's based on has roughly similar events, but very different attitudes/thought processes of people. (I am being somewhat circumspect to avoid spoilers.)