We think of match as an operator that takes two graph-like structures (e.g. database schemas or ontologies) and produces a mapping between elements of the two graphs that correspond semantically to each other. The goal of this paper is to propose a new approach to matching, called semantic matching. As its name indicates, in semantic matching the key intuition is to exploit the model-theoretic information, which is codified in the nodes and the structure of graphs. The contributions of this paper are (i) a rational reconstruction of the major matching problems and their articulation in terms of the more generic problem of matching graphs, (ii) the identification of semantic matching as a new approach for performing generic matching and (iii) a proposal for implementing semantic matching by testing propositional satisfiability.