What with the books and songs thread already going, someone had to start this one.

I'll stick to the films that most influence me the last decade or so:

Before Night Falls
Trois couleurs: Bleu (Three Colors Trilogy: Blue)
La Lengua de las mariposas (Butterfly's Tongue)
Dead Man Walking
Dancer in the Dark (as mentioned in the thread)
One True Thing
Down Came a Blackbird (this was actually made for TV--Raul Julia's last film)
Bamboozled
Four Little Girls

That should get folks started.