Hydraulic valve actuation systems and methods to provide variable lift for one or more engine air valves by way of a variable position hard stop. Various embodiments are disclosed, including embodiments controlling lift by providing a choice of two different fixed stops, three different fixed stops, stops continuously variable throughout a range of lifts, and a fixed stop and stops continuously variable throughout a range of lifts. The valves controlled by a variable position hard stop may be a single engine intake or exhaust valve, or multiple valves of any number, and of either intake or exhaust valves or both, or of one intake or one exhaust valve for one or more cylinders in engines having more than one intake or exhaust valve per cylinder. Dashpot deceleration of engine valve velocity on opening to the hard stop and on engine valve closure is disclosed, as are other aspects and embodiments of the invention.