A scanning tunneling microscope capable of operating at low temperatures,T=400 mK, and in high magnetic fields,B=8 T, is described. Accompanying electronics, under the control of an IBM PC/AT, provide routine spatially resolved spectroscopy, allowing characterization of properties such as superconducting energy gaps and local density of states on the surface. Data are presented to illustrate the usefulness of spatially resolved spectroscopy at low temperatures and in high magnetic fields.