A system for variably actuating valves by means of cams for internal combustion engines has the following type of construction: A camshaft is rotationally supported in the cylinder head and has a fixed position in relation to the valves and their associated stroke-transmitting arrangements. A stroke-transmitting arrangement guided in a fixed position is associated with each of the valves arranged in the cylinder head and closes by means of spring force. A variable position element for adjusting the stroke of the valve is guided in a stationary manner in the cylinder head. An intermediate element is supported and pivotably guided on the variable position element. A cam of the camshaft is engaged by the intermediate element and is supported in a stationary manner in the cylinder head, as well as the stroke-transmitting arrangement associated with the valve.