We investigate what possessives mean by examining a wide range of English examples, pre-and postnominal, quantified and nonquantified, to arrive at general, systematic truth conditions for them. In the process, we delineate a rich class of paradigmatic possessives having crosslinguistic interest, exploiting characteristic semantic properties. One is that all involve (implicit or explicit) quantification over possessed entities. Another is that this quantification always carries existential import, even when the quantifier over possessed entities itself does not. We show that this property, termed possessive existential import, is intimately related to the notion of narrowing (Barker 1995). Narrowing has implications for compositionally analyzing possessives’ meaning. We apply the proposed semantics to the issues of the definiteness of possessives, negation of possessives, partitives and prenominal possessives, postnominal possessives and complements of relational nouns, freedom of the possessive relation, and the semantic relationship between pre- and postnominal possessives.