In this dissertation, a new semi analytical technique has been developed to determine the transformed solutions of non-Newtonian fluids subject to different circumstances involving fractional derivatives. The unsteady motion of viscoelastic fluids, such as Walters’-B fluid, Maxwell fluid, Oldroyd-B fluid and Burgers’ fluid involving fractional derivatives has been discussed with the developed technique. This semi analytical technique has less computational effort and time cost compared to other existing schemes in the literature. In Chapter 2, Caputo-Fabrizio fractional derivative have been developed to study the heat and mass transfer of free convective motion of Walters’-B fluid through an infinite porous vertical plate in the presence of magnetic field. In Chapter 3, the approximate solutions for velocity field and shear stress of a Maxwell fluid using Caputo fractional derivatives have been developed. In Chapter 4 and 5, Caputo fractional derivative has been developed to study velocity field and shear stress in an infinite long circular cylinder subject to an Oldroyd-B fluid and Burgers’ fluid, respectively.