An axe kick is characterized by the straightened leg coming down on an opponent, like the blade of an axe. The starting phase involves the foot being moved in an arc up and forward, like a crescent kick. The arc motion is stopped, and the attacking foot is brought down to strike the target from above, in imitation of an axe. The arc can be performed in either an inward (counter-clockwise) or outward (clockwise) fashion. In some styles, this is known as a downward kick. Axe kicks must be practiced carefully because they can very easily be used to accidentally injure one's sparring partner.
The final move done to Kurt Russel in the movie Death Proof was an axekick to the face.