Originally airing in 1998-1999, Cowboy Bebop is widely regarded as an essential part of anime canon, with very good reason. The story of a group of down-on-their-luck bounty hunters in a gritty version of the future is a mixture of hilarity and tragedy, and worth a watch from almost anyone.
The show begins with Spike Spiegel and his partner, both bounty hunters—dubbed cowboys—on their spaceship, the Bebop (hence the title of the show), in their perpetual search for a new bounty to collect to keep their ship running and food on the table. Spike prophetically declares that he hates working with women, kids, and animals. Over the next handful of episodes, the crew expands by one genetically engineered, hyper-intelligent dog, one child prodigy hacker, and one woman with a lot of debt to pay off and not a lot of moral quandaries about how to get money to pay it. All of the characters have rich histories that color all of their actions, requiring multiple viewings to catch the minor details.
While the animation appears a bit dated, and the dubbing, while well acted, is not well matched with the art, Cowboy Bebop still has one of the best soundtracks ever created.
I always suggest that you try the first five episodes before forming an opinion. If you're not hooked by the end of "Ballad of Fallen Angels" then this is probably not the show for you. If you are, hang on for a wild ride.
See you later, space cowboy...