 We discuss presupposition, the phenomenon whereby speakers mark linguistically information as being taken for granted, rather than being part of the main propositional content of a speech act. Expressions and constructions carrying presuppositions are called “presupposition triggers”, forming a large class including definites and factive verbs. The article first introduces a sample of triggers, the basic properties of presuppositions such as projection and cancellability, and the diagnostic tests used to identify them. The reader is then introduced to major models of presupposition from the last 50 years, separated into three classes: Frege-Strawson derived semantic models, pragmatic models such as that offered by Stalnaker, and dynamic models. Finally we discuss some of the main current issues in presupposition theory. These involve accommodation, which occurs when a hearer’s knowledge state is adjusted to meet the speaker’s presuppositions; presupposition failure, which occurs when a presupposition is (known to be) false; the interaction between presuppositions and attitudes; and variability in the behavior of triggers and their presuppositions.