A detailed calculation of the surface impedance of superconductors is given based on the general theory of the anomalous skin effect in normal and superconducting metals given by Mattis and Bardeen. It is found that there are large corrections to the extreme anomalous limit value of the superconducting to normal surface resistance ratio; corrections to the surface reactance ratio are much smaller. The theory is compared with recent experiments on the surface impedance of aluminum and of tin. It is found that the theory gives satisfactory agreement with experimental data on the surface impedance, both in absolute value and in its temperature and frequency dependence over a wide range of temperatures and frequencies.