An electric power steering apparatus includes a transmission mechanism for transmitting an output from an electric motor for steering assist to a steering mechanism. The transmission mechanism includes a driving gear, an intermediate gear and a driven gear. An eccentric mechanism for eccentrically locating the intermediate gear is provided in order that a center axis of the intermediate gear is moved toward at least one of the axes of the driving gear and the driven gear while maintaining a parallel relation. The eccentric mechanism includes an intermediate-gear support shaft having a center axis. The intermediate-gear support shaft is supported by first and second circle holes in a manner to be rotatable about the center axis thereof for adjustment of a rotational position thereof, the first and second circle holes have their center axes aligned coaxially with each other. The intermediate gear is rotatably supported by an eccentric portion provided at an outer periphery of the intermediate-gear support shaft. The eccentric portion includes a cylindrical surface eccentric relative to the center axis of the intermediate-gear support shaft.