This pioneering study combines insights from philosophy and linguistics to develop a novel framework for theorizing about linguistic meaning and the role of context in interpretation. A key innovation is to introduce explicit representations of context - assignment variables - in the syntax and semantics of natural language. The proposed theory systematizes a spectrum of 'shifting' phenomena in which the context relevant for interpreting certain expressions depends on features of the linguistic environment. Central applications include local and non-local contextual dependencies with quantifiers, attitude ascriptions, conditionals, questions, and relativization. The result is an innovative philosophically informed compositional semantics compatible with the truth-conditional paradigm. At the forefront of contemporary interdisciplinary research into meaning and communication, Semantics with Assignment Variables is essential reading for researchers and students in a diverse range of fields.