The purpose of this thesis is to present the numerical study of some boundary layer flow and heat transfer problems related to the channel with stretching/shrinking walls, orthogonally moving disks, stretching cylinder, vertical square duct, the stagnation point flow and the pulsatile flow. The study includes the self-similar problems and also the ones for which no similarity solution exists. For the problems of first type, a similarity transformation is used to convert the governing Navier-Stokes equations into a set of nonlinear third or fourth order ODEs. The ones of third order are solved by using the order-reduction technique whereas the later are solved by employing quasi-linearization or pseudo transient approach in which the time plays the role of an iteration parameter until the convergence is reached. In case of second type problems, we have used the three step explicit Runge-Kutta method for the numerical study of the unsteady pulsatile flow of a biofluid in the channel, whereas the numerical simulation in a vertical duct has been carried out by employing the Spectral method and the finite difference method. We have studied the flow and thermal properties of not only the Newtonian fluid but also the biofluid (blood), micropolar and Nano fluids as well. The effects of the governing parameters on different aspects of the problems are discussed through tables and figures.