A manga series by Yukito Kishiro. It was also made into a two volume anime OAV series. The protagonist of the series is a cyborg with amnesia who finds fighting the only way to remember her past.
In the United States, the anime is available from A.D. Vision as Battle Angel. Other than the changed title, all other names are translated from the Japanese and so the heroine is Gally. The anime has a dark, violent style which goes well with the realistic writing. The two episodes each cover around a volume of the manga, with appreciable changes. Definitely a suggested rental.
The manga is far better than the anime, however, and the rest of this node will be dealing with the manga. Yukito's art starts out decently and quickly increases in skill and beauty. The storyline has time to proceed and grow throughout the volumes, and the writing is something quite beautiful in the later ones. The characters stick in your mind, and Yukito's not afraid to treat them as real people. There's a lot of interesting details in the series as well - motorball, orbital elevators, and other impressive touches. Some strange references too. An impressive fandom grew up around the manga, complete with fanfics and kooks. Viz published the American edition of the manga, and for whatever reason chose to substantially change the names of people and places, titling the series as Battle Angel Alita. It's still a wonderful work of art.
Apparently, there's a game based on it for the Playstation, but I've never heard much about it.
Battle Angel Alita Volumes:
Battle Angel Alita
Tears of an Angel
Angel of Victory
Angel of Redemption
Angel of Death
Angel of Chaos
Battle Angel Alita Characters
Alita/Gally/Spoiler Name Hidden
Doctor Daisuke Ido
Doctor Desty Nova
Battle Angel Alita Places, Terms and Things