An electric power steering apparatus comprises a rack shaft movable in the axial direction. The rack shaft is meshed with a pinion rotated in response to the operation of a steering member. The rotation of the output shaft in an electric motor is decelerated by first and second gears which are meshed with each other. A motion conversion mechanism for converting the rotation of the second gear into the axial movement of the rack shaft comprises a rotating cylinder rotated in synchronization with the second gear and a screw shaft. The screw shaft is integrally provided coaxially with the rack shaft and is screwed into the rotating cylinder through a rolling member. A screw groove in the screw shaft and the teeth in the first gear are twisted in the same direction.