My favorite romantic movie ever is Pierrot Le Fou.
Sorry, there's just nothing more romantic than a guy leaving his wife and boring bourgeois lifestyle to be with a poor babysitter; as the couple becomes increasingly caught up in an existential Bonnie and Clyde-esque existence, they battle gangsters; blow up cars filled with tons of money; drive into the ocean; karate chop gas station attendants; and forsake everything to live on the beach, with their only concern eventually becoming their own roles as characters in a pop-cultural landscape. It's a genuinely touching and sweet film.
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Also, Kitano's films cause me to tear up, especially his more romantic roles. I mean, he's the best director we've ever had; no director can juggle violence, absurdist humor, and fragile romance with such restraint, control, and beauty. Achilles and the Tortoise is simply my favorite film of the past 10 years -- a story about a failed painter whose wife sticks by his side even as he goes off the deep end. Hana-Bi's story of a cop who borrows money from the yakuza -- and robs a bank to pay them back -- in order to pay for his wife's (presumably) cancer treatments genuinely makes me cry. But I think his ultimate romance statement is

*sniff*
A SCENE AT THE SEA gets me. I'm going to watch it after getting off here. Sniff.
Another great one..

I'M A CYBORG BUT THAT'S OKAY.
This one
really affected me. What a beautiful, unconventional romantic tale.
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I think the absolute MASTER of "romantic" films is probably Wong Kar wai though.
Both stories in Chungking Express are amazing. Fallen Angels is amazing, and depressing. In the Mood for Love.. wow, he's jsut the best.
I can probably think of a thousand great Japanese "romance" movies and a couple hundred French ones, but I can't think of a single American one I really like, except perhaps that David Bowie/patricia aquette movie "The Linguini Inident".