As background for appreciating the still-unsolved problems of monovalent anion selectivity, we summarize the facts and intepretations that seem reasonably well established. In section II we saw that specific effects of monovalent anions on biological and physical systems define qualitative patterns, in that only certain sequences of anion effects are observed. For example, the 4 halides can be permitted on paper as 4! = 24 sequences, yet only 5 of these sequences have been observed in nature as potency sequences. In addition, there are quantitative regularities in anion potency that permit the construction of so-called empirical selectivity isotherms (Figs. 4 and 13). That is, a given potency sequence is found to be associated with only a certain modest range of selectivity ratios. The sequences and isotherms apply to effects with a nonequilibrium component (e.g., permeability and conductance sequences) as well as to purely equilibrium effects. Since students of cation selectivity have had difficulty accepting this conclusion, we discuss the reasons why it is not as paradoxical as it at first seems. In sections III and IV we develop four theoretical models to account for the observed anion potency sequences as sequences of equilibrium binding energies. Two of these models involve calculation of electrostatic binding energies between anions and monopolar or dipolar cationic sites, assuming anions as well as sites to be rigid and nonpolarizable. The other two models use thermochemically measured binding energies between anions and thealkali cations or occasionally alkaline-earth cations, which in fact approximate rigid, nonpolarizable spheres. All four models consider the anion selectivity pattern of a given cationic site to be determined by anion differences in the balance between hydration energies and ion-site binding energies. Site differences in anion selectivity pattern are attributed to site differences in radius, charge, coordination number, or dipole length. These models succeed in predicting all five observed selectivity sequences of the halides. The models predict in addition the possible existence of two further halide sequences that arise from very strong sites and that have not yet been observed in nature. For polyatomic anions the predictions agree approximately but not completely with observations. Thyroidlike systems, which greatly prefer iodide over other halides,re interpreted as having the weakest sites. Site hydration is predicted to affect the magnitude but not the sequence of potency ratios. For the thyroidlike systems, observed potency ratios are smaller than would be expected if anions were completely dehydrated at biological sites.