An accessory device driving apparatus is disposed between an internal combustion engine and an accessory device of a vehicle. The driving apparatus is constructed as a coaxial dual rotor-type electric motor/generator that has two electromagnetically induction-coupled rotors to operate as a motor and a generator. One rotor is coupled with the engine to rotate at a speed higher than that of the engine, and the other rotor is coupled with the accessory device to rotate at a speed higher than that of the accessory device. The rotors are restricted from rotating in a reverse direction opposite a direction of forward rotation of the engine by one-way clutches, respectively.