A torsional vibration attenuation apparatus comprising: an outer peripheral cylindrical member supporting a first power transmission unit (17, 18a, 18b) and having an inner peripheral portion formed with a first screw portion; an inner peripheral cylindrical member supporting a second power transmission unit and provided in the outer peripheral cylindrical member; an intermediate cylindrical member provided between the inner peripheral portion of the outer peripheral cylindrical member and the outer peripheral portion of the inner peripheral cylindrical member and having an outer peripheral portion formed with a second screw portion to be screwed with the first screw portion and an inner peripheral portion splined with the outer peripheral portion of the cylindrical inner peripheral portion, the intermediate cylindrical member being axially movable along the outer peripheral cylindrical member; a first resilient member intervening between the inner peripheral portion of the axial one end portion of the outer peripheral cylindrical member and the outer peripheral portion of the axial one end portion of the intermediate cylindrical member; a second resilient member intervening between the inner peripheral portion of the axial other end portion of the outer peripheral cylindrical member and the outer peripheral portion of the axial other end portion of the intermediate cylindrical member; characterized by further comprising a plurality of ball members intervening between the first screw portion and the second screw portion, wherein both axial end portions of the inner peripheral cylindrical member are formed to have respective projecting portions projecting axially outwardly from the axially both end portions of the outer peripheral cylindrical member, and wherein the torsional vibration attenuation apparatus further comprises: a hysteresis mechanism (30, 31, 32, 33) provided between the inner peripheral cylindrical member and the outer peripheral cylindrical member, the hysteresis mechanism (30, 31, 32, 33) including an urging member and a friction member (32, 33), the urging member being provided between one of the projection portions of the inner peripheral cylindrical member and the outer peripheral portion of one of the axial ends of the outer peripheral cylindrical member to urge the outer peripheral cylindrical member toward the axial other ends of the inner peripheral cylindrical member, and the friction member (32, 33) being provided between the other of the projection portions of the inner peripheral cylindrical member and the outer peripheral portion of the other of the axial ends of the outer peripheral cylindrical member to generate friction force. an outer peripheral cylindrical member supporting a first power transmission unit (17, 18a, 18b) and having an inner peripheral portion formed with a first screw portion; an inner peripheral cylindrical member supporting a second power transmission unit and provided in the outer peripheral cylindrical member; an intermediate cylindrical member provided between the inner peripheral portion of the outer peripheral cylindrical member and the outer peripheral portion of the inner peripheral cylindrical member and having an outer peripheral portion formed with a second screw portion to be screwed with the first screw portion and an inner peripheral portion splined with the outer peripheral portion of the cylindrical inner peripheral portion, the intermediate cylindrical member being axially movable along the outer peripheral cylindrical member; a first resilient member intervening between the inner peripheral portion of the axial one end portion of the outer peripheral cylindrical member and the outer peripheral portion of the axial one end portion of the intermediate cylindrical member; a second resilient member intervening between the inner peripheral portion of the axial other end portion of the outer peripheral cylindrical member and the outer peripheral portion of the axial other end portion of the intermediate cylindrical member; characterized by further comprising a plurality of ball members intervening between the first screw portion and the second screw portion,