A new three-dimensional tunneling theory is introduced for interpreting scanning tunneling spectroscopy (STS) images. By expanding the asymptotic wave function of the acting atom in terms of complete sets of eigen functions in spherical coordinates and parabolic coordinates, a derivative rule is derived, which can be applied to interpret scanning tunneling microscopy and STS images immediately. The relation between the observed dynamic conductance and the density of states of the sample is shown in conjunction with a linear bias-distortion correction. © 1988, American Vacuum Society. All rights reserved.