We present a theory for tunneling between a real surface and a model probe tip, applicable to the recently developed "scanning tunneling microscope. " The tu,nneling current is found to be propor- tional to the local density of states of the surface, at the position of the tip. The effective lateral resolution is related to the tip radius 8 and the vacuum gap distance d approximately as [(2 A)(R+d)]'~2. The theory is applied to the 2&&1 and 3&(1 reconstructions of Au(110); results for the respective corrugation ainplitudes and for the gap distance are all in excellent agreement with 0 experimental results of Binnig et al. if a 9-A tip radius is assumed. In addition, a convenient ap- proximate calculational method based on atom superposition is tested; it gives reasonable agreement with the self-consistent calculation and with experiment for Au{110). This method is used to test the structure sensitivity of the microscope. We conclude that for the Au{110) measurements the experi- mental "image" is relatively insensitive to the positions of atoms beyond the first atomic layer. Fi- nally, tunneling to semiconductor surfaces is considered. Calculations for GaAs{110) illustrate in- teresting qualitative differences from tunneling to metal surfaces.