Cowboy Bebop...
...what a series!
Of course, the anime conventions are totally Japanese, and so is the backstory -- yakusa, and all that. But the drawing is pristine, the character development is excellent, the English dub is very good (including voice actors with wonderful credits), and the music is first-rate.
The movie takes place before the end of the series, but the trailer and clips give some idea what the series looks like. Here.