This book provides a clear and accessible introduction to formal, and especially Montague, semantics within a linguistic framework. It presupposes no previous background in logic, but takes the student step-by-step from simple predicate/argument structures and their interpretation through to Montague's intensional logic. It covers all the major aspects, including set theory, propositional logic, type theory, lambda abstraction, traditional and generalised quantifiers, inference, tense and aspect, possible worlds semantics, and intensionality. Throughout the emphasis is on the use of logical tools for linguistic semantics, rather than on purely logical topics, and the introductory chapter situatesformal semantics within the general framework of linguistic semantics. It assumes some basic knowledge of linguistics but aims to be as non-technical as possible within a technical subject.