This squad of mercenaries has a fairly mixed record; they usually seem to be living hand-to-mouth, despite having been good enough to take down a Prince of Amber, Shaft. It might also, though, be pointed out that they failed to hold him long enough to actually collect the bounty...
They were also less than successful in their attempts to capture Moriya and Davros for House Darios, an operation they engaged in only because Darios was holding Julia hostage. After Jet, Fei and Edward were captured and the full story emerged, a joint operation between the Bebop and the Amberite diplomatic mission led by Miki rescued Julia from Dario's clutches.
They were known to travel shadow in a vehicle known as the Cowboy Bebop (This is what happens when you let Radical Edward name your vehicle, most likely), driven by some sort of Logrus Engine; it was self-destructed by Edward and abandoned when House Darios's giant space whale tried to eat it during the rescue operation. They do mercenary work and bounty-hunting, and are believed to have some degree of ethics (They won't do assassination, for example).
With Julia free, their ship destroyed (though Edward plans to rebuild it) and two major Houses of Chaos unhappy with them, the Bebop have now gone to the hopefully more welcoming clime of Amber to work for Nanami.
Spike is believed to have once been part of some organized crime family, but left it for reasons unknown. Spike is generally thought to be the best fighter and definitely the best shot of the four. He is undoubtedly relieved that Julia is now safe, but doesn't really seem the type to show much surface emotion.
Once a member of the elite Black House Guards, Jet left for reasons unknown, though he recieved an honorable discharge. Not so swift as Spike, he is quite strong. Jet lost an arm to a Primal Chaos attack, and now has a magical limb.
A noted con-woman, Faye is a swift talker and noted shifter, who does much of the group's undercover work.
Often rumored to be the lovechild of the Serpent and Forrest Gump, Radical Edward is the ever-so-slightly deranged and childlike logrus master and artificer of the group. She rarely leaves the Bebop on missions, though her inventions sometimes do. Edward is going to have to build another ship for the Bebop crew, after ordering the old one to self-destruct as it was being boarded by Darios troops during the rescue of Julia.
He's their dog and mascot. Rumored by some to be a powerful shifter trapped in dog form by a shapeshifting accident. Rumored by others to just be a mutt.
Spike's lover. An expert pilot, but not a very capable fighter. Held hostage by House Darios to force the Bebop to pursue Moriya and Davros, she has now been rescued by the combined efforts of the Bebop and the party of Amberites currently in Chaos.