This paper is concerned with the peristaltic transport of an incompressible non-Newtonian Fluid in an elastic tube. Here the flow is due to three different peristaltic waves and two different types of elastic tube. The constitution of blood suggests a non-Newtonian Fluid model and it demands the applicability of yield stress Fluid model. Among the available yield stress Fluid models for blood, the non-Newtonian Casson Fluid is preferred. The Casson Fluid model describes the flow characteristics of blood accurately at low shear rates and when it flows through small blood vessels. Long wavelength approximation is used to linearize the governing equations. The effect of peristalsis and non-Newtonian nature of blood on velocity, plug flow velocity, wall shear stress and the flux flow rate are derived. The flux is determined as a function of inlet, outlet, external pressures, yield stress, amplitude ratio, and the elastic properties of the tube. Furthermore, it is observed that, the yield stress, peristaltic wave, and the elastic parameters have strong effects on the flux of the non-Newtonian Fluid, namely, blood. One of the important observation is that the flux is more when the tension relation is an exponential curve rather than that of a fifth degree polynomial. Further, in the absence of peristalsis and when the yield stress tends to zero our results agree with the results of Rubinow and Keller (1972). This study has significance in understanding peristaltic transport of blood in small blood vessels of living organisms.