Introduction: Since its first appearance in van Fraassen’s semantics for free logic, the notion of a supervaluation has been regarded by many as a powerful tool for dealing with truth-value gaps and, more generally, with phenomena involving semantic indeterminacy, partiality, deficiency of meaning. Unlike three-valued semantics, whose truth-functional character incluced a proliferation of competing variants, supervaluational semantics appeared to offer a uniform way of keeping these phenomena under control; and work in the following decades has supported this expectation with a fair deal of significant developments. These include, among others, applications to such diverse domains as quantum logic, temporal logic, vagueness, presuppositions, sortal incorrectness, or the semantic paradoxes. In light of this prosperity, it is remarkable that supervaluation-like methods have not registered a comparable fortune (if any) in connection with truth-value gluts, i.e., more generally, with phenomena involving semantic overdeterminacy or inconsistency as opposed to indeterminacy and incompleteness. Part of the explanation is, of course, to be found in the greater hostility that such forms of semantic anomaly have registered on the whole. However, truth-functional methods have been applied to semantics with gluts, or with gaps and gluts alike. For instance, four-valued generalizations of Kleene’s (1938) three-valued matrices have become rather popular among those who view gaps and gluts as two sides of the same coin, two complementary ways in which a semantics may fail to be uniquely determined. One wonders, then, whether and to what extent similar generalizations are available ‒ at least in principle ‒ to the friends of supervaluationism. The purpose of this paper is to show that one can actually go quite far in that direction. There is no intrinsic difficulty in applying the supervaluational insight to interpret a language in the presence of semantic inconsistency. More generally, there is no intrinsic difficulty in generalizing the concept of a supervaluation so as to deal with both kinds of semantic anomaly ‒ gaps and gluts. In fact there are many options available; and in all cases the distinguishing features of van Fraassen’s original technique can be preserved: (i) the general notions of satisfiability and refutability are non-classical, but (ii) logical truths and logical falsehoods (and more generally entailment relations between sentences) retain their classical status.