There’s a French animated film that I love; A Town Called Panic.
They play it a lot on cable. Very surreal and hilarious.
Most of mine have been listed.
La Fem Nikita
The Killer
Kimagure Orange Road: I Want to Return to That Day ( requires knowing the previous OAV’s at least )
Meflin
Most of mine have been listed (I loved Rust and Bone), but I haven’t seen the 2002 winner for Best Foreign Language Film, Nowhere in Africa, which is probably my favorite foreign language film (in German).
No Man’s Land-Bosnian War drama. 2001 Oscar winner Best Foreign Language film.
Man Bites Dog-French mockumentary about a film crew following and profiling a serial killer.
The Secret In Their Eyes
Walkabout
Strangers In Good Company
La Cage Aux Folles
Some great movies mentioned, but this one was my favorite for a good long while:
**Mitt Liv Som Hund (My Life As A Dog) ** - a Swedish coming of age story, about a curious and imaginative young boy who is forced to live with relatives as his mother goes through health problems. The film has noticeably bad editing at time (in terms of cuts), but is otherwise one of the best unflinching depictions of a kid and his world and thoughts I’ve seen. I’m making it sound heavy, but it’s really quite funny too.
A superb satire of media manipulation, very sick and very funny and 20 years before its time.
I’d add Hidden (Caché) , though I had to look up what happened at the end. In this someone is videotaping a family, and posting the tape through the door.
I’d also ask if anyone understands Certified Copy, then please tell me, but I did appreciate its strange ways.
And a rare soviet film about the hell that was the east after the Germans invaded in WWII: Come and See. Astounding.
By this criteria, Once.
I loved Whale Rider, too, but it isn’t a foreign film (to me, anyhow),
Burnt By The Sun a Russian film (1994)
The Square, an Australian film about married lovers plotting to kill her husband. I’ve watched it twice, and there’s not a dull moment.
Ditto The Secret in Their Eyes, although the rape/murder scene is very difficult to watch.
As it should be. A rare example (IMO) of non-gratuitous horror. As much as I love this movie, during those scenes I yank myself out of the movie - “they’re actors” - and then jump back in.
I go back and forth on Krzysztof Kieslowski’s Blue and Red as to which is my favorite of the two. My favorite is whichever I’m watching at the time, I guess.
The Orphanage (El Orfanato in Spanish) - excellent, non-gory horror flick from Spain.
The Full Monty, I still remember it after all these years. Funny, yet poignant film.
Le Cop (Le Ripoux) and Le Cop II…
Wonderfully funny French ‘buddy’ films about two cops. The hard-bitten cynical one meets the ‘straight out of college’ idealist. Well worth a watch…
Requiem fur Dominik…
A Romanian film about the fall of Communism. Acting mixed with contemporary footage. Very powerful.
I once saw a South American film that started in the middle, half way through had the end credits, swiftly followed by the opening credits, and then ended in the middle (where it started). It was about a woman who was trying to find out who was throwing eggs at her house. I know it sounds absolutely bonkers, but it worked.
Anyone remember it?
Me neither, but other posters had listed English-language non-US films and I take any chance to recommend Whale Rider. 
Here’s everything on the list of my 100 favorite films that probably would be considered non-U.S. movies:
Amarcord (1974, Italy, Federico Fellini)
Chungking Express (1994, Hong Kong, dir. Wong Kar-Wai)
Excalibur (1981, U.K., dir. John Boorman)
Fantastic Planet (1973, France, dir. Rene Laloux)
Four Weddings and a Funeral (1994, U.K., dir. Mike Newell)
Jules and Jim (1961, France, dir. Francois Truffaut)
La Jetée (1962, France, dir. Chris Marker)
La Strada (1954, Italy, dir. Federico Fellini)
M (1931, Germany, dir. Fritz Lang)
Metropolis (1926, Germany, dir. Fritz Lang)
My Brilliant Career (1979, Australia, dir. Gillian Armstrong)
The Road Warrior (1981, Australia, dir. George Miller)
Seven Beauties (1976, Italy, dir. Lina Wertmuller)
The Seven Samurai (1954, Japan, dir. Akira Kurosawa)
The Third Man (1949, U.K., dir. Carol Reed)
The Tree of Wooden Clogs (1978, Italy, dir. Ermanno Olmi)
The Year of Living Dangerously (1983, Australia, dir. Peter Weir)
I can’t explain it either, unfortunately, but I still thought it was pretty good. It’s like a puzzle that I wasn’t able to figure out, but I think it’s a credit to the movie that I enjoyed it even thought I wasn’t able to make everything fit together in the end. Plus, Juliette Binoche’s performance is wonderful.
M.*
*sometimes it is difficult to avoid whistling “The Hall Of The Mountain King” when in a public place where children are misbehaving.