Peristaltic pumping by wave-like muscular contractions is a fundamental biomechanical mechanism for material transport and is used in the intestine, oviduct and ureter. While peristaltic pumping of Newtonian fluids is well understood, in many important applications ( e.g. fluid dynamics of reproduction) the fluids exhibit non-Newtonian behavior. I will present numerical simulations of the Oldroyd-B equations investigating such non-Newtonian effects over a range of Weissenburg numbers, peristalsis wavelengths and amplitudes. A similar problem computationally is the locomotion of a swimmer in an Oldroyd-B fluid. I will present numerical results that reproduce the classical asymptotic predictions of G.I. Taylor for a periodic swimming sheet in a Stokesian fluid as well as more recent predictions by E. Lauga for an Oldroyd-B fluid. I will also discuss interesting non-Newtonian behavior and its effect on locomotion strategies. I will conclude with a brief advertisement/introduction for an AML seminar (March 29) I am giving on virtual surgery with the NYU medical school.