Scanning tunneling microscopy (STM) has been used for many atomic scale observations of metal and semiconductor surfaces. The fundamental principle of the microscope involves the tunneling of evanescent electrons through a 10Å gap between a sharp tip and a reasonably conductive sample at energies in the eV range. Lateral and vertical resolution are used to define the minimum detectable width and height of observed features. Theoretical analyses first discussed lateral resolution in idealized cases, and recent work includes more general considerations. In all cases it is concluded that lateral resolution in STM depends upon the spatial profile of electronic states of both the sample and tip at energies near the Fermi level. Vertical resolution is typically limited by mechanical and electronic noise.