The process of on-line generation of ultrapure dialysis fluid is a core prerequisite for the safe execution of modern renal replacement therapies such as on-line hemodiafiltration and high-flux hemodialysis. In these extracorporeal treatments with variable degrees of convection, significant volumes of plasma water are removed and replaced with dialysis fluid, which must occur without causing harm to the patient. Historically, on-line generation of sterile and pyrogen-free physiological substitution fluid by the process of membrane ultrafiltration of fresh dialysis fluid has its origin in hemofiltration, a purely convective therapy. Development of this and later therapies is described in the historical context of a successful effort over decades to overcome the above formidable challenge, which was provided jointly by pioneering clinical investigators and a resourceful dialysis industry.