"Battle Angel" is based on the Japanese Manga series of graphic novels by Yukito Kishiro. The movie combines the first three books, Ido, Alita, and Hugo.
Set in the 26th century, the story takes place 300 years after a societal collapse caused by a major war. But in that society, it's a technological dark age following a pinnacle of achievement far, far beyond where we are right now. It's a post-apocalyptic world from a very high level with cyborg technology as just a way of life. People are augmented a lot as workers and so on, so being a cyborg is not unusual.
The main character is a cyborg. She has an organic human brain, and she looks like she's about fourteen years old. She has a completely artificial body and she's lost her memory - she's found in a wreckage and she's reconstituted by a cyber-surgeon who becomes her surrogate father.