In this book we analyze the heat and mass transfer effects on unsteady free (or mixed) convection flows of a Newtonian (non-Newtonian) fluid past a vertical porous plate under varying initial and boundary conditions. The plate is assumed to move with a constant velocity in the direction of fluid flow. A uniform magnetic field acts perpendicular to the porous surface .The dimensionless governing equations for these investigations are solved analytically using two-term harmonic and non-harmonic functions. In order to get a physical insight into the problems, the behavior of the velocity, temperature, concentration, skin-friction, Nusselet number and Sherwood number are studied and displayed graphically for pertinent parameters.