Magnetic reconnection is a fundamental physical process by which stored magnetic energy may be released. It is already known that different reconnection regimes result from changes in the nature of the plasma inflow towards the reconnection site. In this thesis, we examine both how the outflow region responds to changes both in the inflow and outflow boundary conditions and also how introducing compressibility affects the results. We find that if the inflow is converging, the outflow velocity is least, the width of the outflow region is greatest and the ratio of outflowing thermal to kinetic energy is greatest. Also, there is one free outflow parameter which would naturally be specified by the velocity of plasma leaving the reconnection site. We suggest that reverse currents seen in numerical simulations may result from the specification of an extra boundary condition. In addition, we find that the main effects of including compressibility are: to enhance convergence or divergence of the inflow; to increase the maximum reconnection rate where the inflow is converging; to increase the flow speed near the reconnection site where the inflow is diverging; to give faster, narrower outflow jets; to increase variations between regimes in the energy conversion and to increase the ratio of thermal to kinetic energy in the outflow jet.