Pumping cartridges useful for medical and surgical pumping and infusion procedures. In one aspect, a series of pumping cartridges include cylinders and pistons which have a sealing component including a circumferential, flared sealing flange and can be configured to enable the pumping cartridge to generate high liquid pumping pressures. Many of the pumping cartridges can be configured to be detachable from a pump drive unit and disposable after a single use. Some pistons include or at least partially form a valve and/or are relatively movable with respect to a connected piston rod. Valves include a valve seat and a movable sealing element able to create a seal capable of withstanding differences in fluid pressures thereacross of at least about 1,000 psi. The movable sealing element is concave in shape, and may be curved for improved sealing performance. Methods for pumping or infusing a liquid for delivery to the body of a patient or a surgical or medical instrument.