The present invention generally relates to hemodialysis and similar dialysis systems, including a variety of systems and methods that would make hemodialysis more efficient, easier, and/or more affordable. One aspect of the invention is generally directed to new fluid circuits for fluid flow. According to one aspect, a blood pump is configured to pump blood to a dialyzer of a hemodialysis apparatus, the blood pump comprising a pneumatically actuated or controlled reciprocating diaphragm pump. In an embodiment, the diaphragm of the pump comprises a flexible membrane formed or molded to generally conform to a curved inner wall of a pumping chamber or control chamber of the pump, wherein the diaphragm is pre-formed or molded to have a control side taking a convex shape, so that any elastic tension on the diaphragm is minimized when fully extended into a control chamber of the pump. In another aspect, a system for monitoring the adequacy of blood flow in a blood line of the hemodialysis apparatus allows a controller to suspend dialysate pumping operations if the adequacy of blood flow in the blood line is sub-optimal, and to present information on a display on the quality of blood flow in the blood line.