Potential and current distributions and local energy dissipation due to Joule heating in metal-superconductor junctions have been computed as a function of geometric parameters and interfacial resistance. The primary current distribution and power dissipation are highly nonuniform in the system. The secondary current distribution and power dissipation, however, become more uniform as the interfacial resistance increases. Analysis indicates that zero contact resistance is not a stable situation since the primary distribution leads to local current densities exceeding the critical current density of the superconducting phase near the corner of the junction. Local contact failure might then initiate. A finite contact resistance is necessary for a practical application, and the minimum value of the contact resistance can be estimated from the operating current density (javg) of the device and the critical current density (jcri) of the superconducting phase. To obtain an optimum value of the contact resistance, however, one further has to take into consideration the stability and reliability of the device performance, which is, in turn, directly related to the uniformity of the current distribution and power dissipation, to temperature fluctuation of the superconducting phases brought about by local power dissipation, and to the thermal management of the system. Furthermore, a nonuniform contact resistance layer of appropriate profile can redistribute the current more effectively and more uniformly and hence reduce the total power dissipation in the system for a given jmax/javg ratio obtained by a uniform resistance layer.