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.
In the year 2071, a group of outlaw bounty hunters travels through space pursuing jobs that will earn them a big payday. While some of what they encounter does border on the supernatural, none of the main characters possess any special magical abilities.
They collect bounties through physical clashes, ingenuity, and more - all set to one of the best soundtracks in anime history.
When businessman Rokurou Okajima is kidnapped by the mercenary group Black Lagoon, he's abandoned by his employers and ends up deciding to join the group as a full-fledged member. Black Lagoon's mercenary work is risky business, and the members' reasons for participating in it are engaging because of their realism.
Revy's childhood of police misconduct and prejudice is resonant enough without a supernatural component.
Fuu is a former waitress on the hunt for a samurai who smells like sunflowers. She hires two samurai who originally helped her deal with harassment at work, Jin and Mugen, to accompany her on her journey. Though Jin and Mugen are powerful warriors, none of their abilities defy the limits of human physicality - they've both trained hard in martial arts and swordsmanship.
Most of the problems they encounter on their journey are political or interpersonal, and they don't delve into the world of magic.
Kingdom takes place during the period of China when states clashed. It tells the story of two young slave children, Piao and Xin Li, who dream of someday rising from their stations and becoming a Great General of the Heavens. One day, the two boys are separated when Piao is called to the royal palace. When he returns, he's a shell of his former self - and he has a mission for Xin.
Kingdom is based on actual Chinese history, which offers such a rich storytelling vein that there's no need to infuse it with anything supernatural.