The cleavage faces for zinc blende structure compound semiconductors are the (110) surfaces. Those for wurtzite structure compound semiconductors are the (101̄0) and (112̄0) surfaces. The (100) and (112̄0) surfaces consist of chains of equal numbers of anion and cations, each of which exhibits two surface bonds and one back bond. The (101̄0) surface consists of pairs of anions and cations, each of which exhibits one surface and two back bonds. All of these surfaces relax relative to their bulk geometry with the anion moving outward and cation inward until the cation achieves an approximately planar sp2 coordination. These relaxations approximately conserve bond lengths, although surface and back bonds may be altered by a few percent. Second and deeper layer relaxations are predicted but have been confirmed experimentally only in a few cases. The mechanism for the relaxations is the (activationless) lowering in energy of a completely filled band of anion derived surface states as they change from dangling-bond to surface- and back-bonding states. The relaxations are ‘‘universal’’ in that for a given surface all materials exhibit approximately the same structure when distances are measured in units of the bulk lattice constant. A complete set of theoretical predictions, structure determinations, and surface-state eigenvalue spectra is available for the (110) surfaces of GaAs, InP, ZnSe, and CdTe as well as for CdSe (101̄0). Theoretical predictions of the structure and surface state eigenvalue spectra have been given for all common IIII–V and II–VI binary semiconductors. Experimental structure determinations have been reported for their zinc blende (110) surfaces, the (101̄0) surfaces of ZnO and CdSe, and the (112̄0) surfaces of CdS and CdSe. Both the predictions and measurements confirm the notions of the universality of the surface relaxations and the dominant role of surface topology in determining their character.