he main aim of this work is to study the e↵ects of various aspects such as Hall current, free convection, radiation, heat absorption and porous medium on heat transfer behavior in viscous, incompressible and electrically conducting fluids past a vertical plate in the presence of an externally applied magnetic field. This thesis contains four chapters, the description of which are given as follows: Chapter one consists of important definitions of terms and expressions. The literature review, aims and objectives of this study as well as the research methodology are described. Chapter two presented simultaneous e↵ects of thermal and concentration di↵usions in unsteady magnetohydrodynamic free convection flow past a moving plate maintained at a constant heat flux and embedded in a viscous fluid saturated porous medium. The transport model employed includes the e↵ects of thermal radiation, heat sink, Soret and chemical reaction. The fluid is considered as a gray absorbing-emitting but non-scattering medium and the Rosseland approximation in the energy equations is used to describe the radiative heat flux for optically thick fluid. The dimensionless coupled linear partial differential equations are solved by using Laplace transform technique. Numerical results for the velocity, temperature, concentration as well as the skin friction coe""cient and the rates of heat and mass transfer are shown graphically for di↵erent values of physical parameters involved..