Investigation of flow and heat transfer phenomena in a channel filled partially or fully with porous medium is of significant interest as it has numerous applications in heat exchangers, crude oil extraction, jet printers, petroleum reservoir, filtration mechanism, magnetic refrigerator, rocket propulsion etc. It is observed that the pressure gradient and temperature difference play vital role in the channel and pipe flows. Also, presence of porous medium has a significant effect on heat transfer phenomenon. Theoretical analysis of such flows is helpful to examine more complex problems related to flow and heat transfer phenomena. A lot of research has been conducted in this direction. However, characteristics of heat transfer in composite channel flows are yet to be explored as a few literature is available on this topic. Keeping this in mind, emphasis has been paid on the steady of flow and heat transfer characteristics in channels partially or fully filled with porous medium. Newtonian and non-Newtonian fluid known as Casson fluid have been considered and the impact of presence of magnetic field, unsteadiness and suction/injection on fluid flow and heat transfer are observed. The considered problems are mathematically modelled using momentum and energy equations. In some particular cases, entropy generation effects are also investigated to measure the energy losses during these processes. Different techniques like numerical and analytical are used to solve partial differential equations which govern heat transfer and flow phenomena. Regular perturbation method is used to solve analytically the governing equations whereas finite difference scheme are employed to obtain the numerical solutions. The validity and accuracy level of the solutions have been determined either by comparing both the solutions or calculating the residual error. The analysis carried out in this dissertation reveals that the presence of porous medium decreases the velocity of the fluid within the pipes or composite channels. Also, it is noticed that the flow and heat transfer strongly depend on the viscosity ratios, magnetic field parameter, thermal conductivity ratios, inclination angle and porosity parameter. Moreover, the Newtonian fluid velocity is higher than the Casson fluid velocity for irregular channel with convective boundary conditions. In case of composite channel, temperature enhancement is more in porous region as compared to clear region. Furthermore, presence of porous medium, magnetic field parameter and viscous dissipation effects are the major factor responsible for entropy generation.