This academic paper deals with reviewing theoretical studies on MHD peristaltic transport of the Non-Newtonian as well as Newtonian fluids such as Hyperbolic Tangent fluid, Carreau fluid and Bingham fluid. Here, a wide range of study subjects, concepts, points of view, and mathematical models are presented. All of these studies are focused on Non-Newtonian fluids peristaltic activity. Among  numerous of the Non- Newtonian fluids flows in physiological system, blood pumping mechanics