Absolutely the best anime I've ever seen is Cowboy Bebop. Combining elements of film noir and spaghetti westerns, this series excels for its incredible action sequences, quirky, complex characters, and magnificent score.

Cowboy Bebop is a space western following the adventures of four bounty hunters, set in 2071. Each episode centers around a specific bounty being tracked down, with a few cases running over more than one episode. The series also features a jazz-style score deftly incorporated into the episodes, often developing the themes musically.

Cowboy Bebop is periodically rebroadcasted on Cartoon Network's Adult Swim after being the first anime to be shown on this program in 2001. It's popularity was boosted by a feature length movie (Knocking on Heaven's Door) which was aired in the States in 2005. It should be noted, however, that this is an adult anime with some scenes and references which may not be suitable for children.

Cowboy Bebop's appeal is wide, attracting even people previously disinterested in anime. This is due to technical excellence, interesting characters, and independent episodes. All things considered, Cowboy Bebop is an excellent introduction to anime and a must-see for anyone with an interest in the art form.