An electric power steering apparatus comprises a ball nut rotationally driven by an electric motor through a gear train and having a female thread groove formed in the inner periphery thereof, a rack shaft disposed through the ball nut on the axis thereof and having a male thread groove so formed in its outer periphery as to face to the female thread groove, a ball screw mechanism having a number of circulation balls interposed between the female thread groove and the male thread groove, and a housing for holding the ball nut rotatably through the bearing. An elastic member is interposed between the housing and the ball nut in order to permit the ball nut to make a predetermined displacement.