The investigation carried out in this thesis focuses on the numerical analysis of time-dependent flow and heat transfer of a non-Newtonian Williamson fluid. In the literature, several constitutive equations were calculated which describe the relation between stress and rate of strain for non-Newtonian fluids. We have chosen the constitutive equation as suggested by Williamson for pseudoplastic materials and proposed the model equations to describe the boundary layer Williamson fluid flow. The core focus of this thesis is to study the behavior of different geometries, like, planar stretching sheet, radially stretching sheet, stretching/shrinking sheet, stretching cylinder, expanding/contracting cylinder and static/moving wedge, for flow patterns of nonNewtonian Williamson fluid. We know that when fluid flows over a solid body, such as the hull of a ship or an aircraft, frictional forces retard the motion of the fluid in a thin layer close to the solid body. The development of this thin layer is a major contributor to the flow resistance and is of great importance in many engineering and industrial problems. Therefore, the study of such boundary layer flows of non-Newtonian fluids due to different stretching geometries have gained remarkable aspects in numerous industrial applications. Keeping this in view, the present thesis finds out how the stresses on the bodies are affected and to know the behavior of the Weissenberg number of these stresses in relation to the contribution of zero and infinity shear rate viscosity. Additionally, the non-Newtonian Williamson fluid flow due to different stretching surfaces finds its extensive applications in the area of agriculture, engineering, petroleum industries, geothermal reservoir and geothermal energy extractions. This thesis further presented the characteristics of heat and mass transport of nonNewtonian Williamson fluid. A computational code is developed for the present analysis and it is verified against the available numerical data. Numerical outcomes characterizing the performances of velocity, temperature and concentration distributions are captured through graphical illustrations. The surface drag force, heat and mass transfer rates are also obtained. It is found that the Weissenberg number slow down the fluid motion while it enhances the temperature distribution. Further, it is worth mentioning that an increment in the unsteadiness parameter diminishes the fluid velocity and temperature, respectively. It is interesting to note that the higher the viscosity ratio parameter has a tendency to decrease the skin friction coefficient substantially.