Abstract: In an on-vehicle hybrid drive apparatus including a power-transmission shaft transmitting rotation generated from an engine into a transmission, and a motor-and-generator fitted on the power-transmission shaft and installed between the engine and the transmission, a first friction element is installed on the engine side for coupling the engine with or uncoupling it from the motor-and-generator. A second friction element is installed on the transmission side for coupling the motor-and-generator with or uncoupling it from a transmission output shaft. A rotating damper is installed after the motor-and-generator and disposed in a rotating-motion transmission system ranging from the motor-and-generator to the transmission output shaft. The rotating damper is interleaved in a coaxially abutted shaft portion of a central motor-and-generator shaft and a transmission input shaft.