Battle Angel Alita is the translated English title of Yukito Kishiro's manga series Gunnm. An approximate translation of that is "Gun Dream", and that's certainly a thematic element. Alita explores the distant future where the Earth is in a post-apocalyptic state, almost completely desertified. While technology is extremely advanced, the vast majority of society is disorganized, a network of government contracted bounty-hunters being the only visible arm of the law.

Rescued from a scrap heap by a mechanic, Alita is a cyborg, though before the repairs she's barely more than a head and a fragment of her body. She does not know much about her past, but retains an instinctive knowledge of an extremely powerful fighting technique called Panzer Kunst.

The official English translation is available from Viz.

Background:

The underclass are forced to work for the elite people from Tiphares, a massive floating city where they all live. They get little in exchange, except that which they get through the black-market. The people from Tiphares are identifiable by a mark on their forehead, something they are given as a right of passage when they reach a certain age.

Alita is discovered by Ido, a bounty hunter and doctor/mechanic (the difference isn't entirely clear with cyborgs) laying in a junkpile, discarded by the Tipharians. He is skillful enough to restores her to working condition.