An electric power steering system for vehicles including a driving control circuit (76, 100, 108) for feeding, in consideration of an output signal (S.sub.1, S.sub.2) from a steering torque dectection mechanism (77) for detecting steering torque (Ts) acting on an input shaft (4), a drive signal (Va) to an electric motor (33) for producing auxiliary torque to be supplied to an output shaft (7). The steering system further comprises a detection mechanism (82) for detecting the steering speed (Ns) of the input shaft (4), and the driving control circuit (76, 100, 108) is adapted to feed the drive signal (Va) to the electric motor (33) in consideration of an output signal (S.sub.4, S.sub.5) from the steering speed detection mechanism (82) as well as of the output signal (S.sub.1, S.sub.2) from the steering torque detection mechanism (77). The driving control circuit (76, 100, 108) includes a micro-computer unit (76) for determining by address designation a control signal (T.sub.3, T.sub.4 , T.sub.5) as basis of the drive signal (Va) to the elecric motor (33), depending on the output signal (S.sub.1, S.sub.2) from the steering torque detection mechanism (77) and the output signal (S.sub.4, S.sub.5) from the steering speed detection mechanism (82).