An electric power steering apparatus for converting rotation of a steering assist motor into movement of a steering shaft in the axial direction thereof in accordance with steering operation using a ball screw mechanism wherein a female thread groove formed around an inner circumference of a rotating cylinder rotated around the axis thereof by the motor and a male thread groove formed around an outer circumference of the steering shaft are screwed with each other via a plurality of balls. The portions, where the both thread grooves contact with the balls, are formed in a circular arc shape, respectively, and the curvature of the female thread groove is made smaller than that of the male thread groove. Hence, the ball screw mechanism wherein skew occurrence between the ball and both the thread grooves is prevented and the dimensional errors of both the thread groove are decreased is realized.