A valve operating apparatus, which transmits the rotation of the crankshaft of an automotive engine to the intake and exhaust valves to open and close the valves, can selectively vary the timing to open and close the valves and ther lifts of the valves. The valve operating apparatus includes a camshaft with low- and high-speed cams and low- and high-speed rocker arms, one of which is fixed to and the other rotatably supported on a rocker shaft. The valves can be opened and closed by the low-speed rocker arm. The other rocker arm and the rocker shaft can be engaged and disengaged by an engaging mechanism that is actuated by an actuating mechanism depending on the operating condition of the engine.