The total soft x-ray emission of an electron-bombarded surface exhibits abrupt changes at the threshold potentials for the excitation of core levels of surface atoms. These features represent the excitation probabilities of core states superimposed on a smoothly increasing bremsstrahlung background. The threshold potentials can be sensitively detected by differentiating the current from an x-ray photocathode with respect to sample potential. This technique represents the highest resolution core-level spectroscopy available. We discuss the principles underlying the design and operation of soft x-ray appearance potential spectrometers and review their application to the study of surface composition, electron binding energies, chemical shifts, and band structure.