This thesis aims at understanding and improving results in the field of non-Newtonian fluid mechanics. The primary objective of this work is the mathematical modeling and development of numerical solutions for boundary layer flows involving non-Newtonian Carreau fluid. For this purpose, a two-dimensional boundary layer equations are developed for varying stretching geometries like planer stretching sheet, radially stretching/shrinking sheet, stretching/shrinking cylinder and moving wedge. The governing problems concerning the flow and heat transfer analysis of Carreau fluid over non-linear stretching surfaces have been formulated with the aid of Boussinesq approximations. The modeled partial differential equations (PDEs) are transformed to ordinary differential equations (ODEs) by utilizing suitable transformations. To gain a better insight about the behavior of these flows, numerical solutions are developed with comprehensive and meaningful interpretation. For the numerical simulation, we apply two numerical techniques: Runge-Kutta Fehlberg scheme and the collocated method (bvp4c). There is a great diversity of non-Newtonian fluids flow in chemical engineering and industrial process, like, suspensions, nuclear fuel slurries, emulsions and polymeric solutions, etc. Definitely in depth study of these fluids is an important and challenging task in the engineering field. Therefore, the present thesis demonstrates the mathematical modeling and numerical simulation to the boundary layer flows of a non-Newtonian Carreau fluid. In this work a qualitative study is performed with a few profound observations. The acquired results in this work for the flow and heat transfer characteristics of Carreau fluid are mainly presented in the form of velocity, temperature and concentration profiles. The general trend of velocity profiles is that a higher estimation of the power-law index leads to an enhancement in the fluid velocity as well as momentum boundary layer thickness. However, a quite the opposite is true in case of temperature field. Additionally, the fluid has a thicker boundary layer thickness in case of shear thickening fluid. It is worth highlighting at this end that upon increasing the Weissenberg number the velocity and the momentum boundary layer thickness depreciate in case of shear thinning fluid. One the other hand, in contrast with shear thinning nature an inverse trend is seen for shear thickening fluid. The temperature profiles endorse in the sense that the temperature and thermal boundary layer have the opposite characteristics as that of fluid velocity. In addition, it must be observed that such effects are much prominent in shear thinning fluid as compared to shear thickening fluid.