We examine the onset of superconductivity in the surface region of a metal. Surface effects are particularly important in systems with a short bulk coherence length 0. We show that, to the accuracy of the calculation, the surface transition temperature TcS equals the bulk transition temperature TcB if the electron-electron interaction is of the standard BCS form, i.e., a single attractive square well, extending up to some critical energy 0 much smaller than the Fermi energy F. If one takes into account, in addition, the repulsive part of the interaction extending beyond 0 up to energies of order F, then one may have TcS>TcB in certain cases, although, due to restrictions imposed on the parameter values by various physical conditions, the relative increase of Tc is very small, typically 10-3, at least in the weak coupling limit. However, we also find a considerable gap enhancement, of order 20%, near the surface which could be of interest for critical-current measurements. Therefore we suggest an experimental reexamination of systems with short 0, i.e., superconducting degenerate semiconductors and the new high-Tc oxides in confined geometries where the surface-to-volume ratio is non-negligible.