Hemodialysis and similar dialysis systems including a variety of systems and methods that make hemodialysis more efficient, easier, and/or more affordable, and include new fluid circuits for fluid flow in hemodialysis systems and a reciprocating diaphragm pump for pumping fluids. The reciprocating diaphragm pump includes a flexible diaphragm, a first rigid body having a curved pumping chamber wall, a second rigid body having an opposing curved control chamber wall. The diaphragm is interposed between the pumping chamber wall and the control chamber wall to define a pumping chamber and a control chamber. The diaphragm of the pump has a peripheral bead arranged to locate the diaphragm between the first rigid body and the second rigid body and a diaphragm body having a curved, semi-spheroid or domed shape. The diaphragm is pre-formed or molded so that during a delivery stroke of the pump, the elastic force of the diaphragm resisting its deployment into the pumping chamber prevents a peripheral portion of the diaphragm body from fully contacting the pumping chamber wall.