In just about every action-based anime, you will find characters with magical abilities ranging from telekinesis to ice beams. While these powers can add flavor to a story, they can also make the conflicts feel too easy to resolve. You may be looking for some good action anime without superpowers. If so, there's plenty to choose from. Do you like stories based around nefarious groups? Try Banana Fish, the recent hit that follows teenage Mafioso Ash Lynx as he seeks to uncover the mysteries of the titular substance. Prefer a more historical approach? Try Kingdom, an anime about the period in ancient China when states clashed.
Note that anime with no special abilities aren't necessarily free from supernatural elements. High School of the Dead is about a zombie outbreak, which is certainly a supernatural event. However, the characters have no supernatural powers to help them deal with it - they just have regular old human ingenuity.