Blood+(2005)

Blood+

Saya, the main character, lost her memory and has been living with her adopted family. Her everyday life seems quite normal, except that she eats a lot more than she actually needs. Then one day she meets a guy named Hagi in a park. She is told that he has been searching for her, for a long time. Since then, her normal life has changed completely. (This manga series is proceeding quite a bit differently from the anime)
A high school girl named Saya can not remember anything from her past. A handsome young man, Hagi, appears and something seems familiar about him. This girl is more than human with an ancient and mysterious past. She will hunt down and kill the monsters called Chiropterans.