A rotary electric machine such as an alternator or a motor has an engine-cooling water-pump in an integral unit. The rotary electric machine is composed of a frame secured to a front surface of an engine, an electric machine unit , a pump unit and a driving unit. The frame has a wall portion and a water passage formed on a rear surface of the wall portion and connected between the engine cooling system and the engine. The electric machine unit has a stator disposed in thermal contact with a front surface of the wall portion and a rotor fixed to a shaft of the driving unit. The pump unit is fixed to the frame and has a pump impeller disposed in the water passage to take cooling water from the cooling system to the engine through the cooling water passage. The driving member includes a pulley driven by an engine and a shaft supported by the wall portion through a bearing to drive the rotor and the impeller.