This show is essentially a sequel to Batman: The Animated Series set 40 years in the future. At first it takes a while to accept Terry as Batman, but by the end of the show, he is Batman. The first season was masterful, featuring the return of a couple foes from the original including Mr Freeze and Bane. For the most part though, there is a new cast of baddies icluding Blight, Inque, Spellbinder, the Royal Flush Gang, and Shriek. Season 2 lags a bit, but season 3 finishes strong with an episode featuring the futur Justice League and Superman. Its amazing how good the show turned out, and how true to Batman this show turned out based on a premise that could have been a disaster.
If you like this show, also check out the DTV Return of the Joker, it is one of the best Batman's stroies on film. Also, be advised that the episode Epilogue of the series Justice League Unlimited serves as Batman Beyond's true finale.