Recent developments in the theory of interpretation of natural language suggest that a strict separation of the study of meaning (semantics) and its use (pragmatics) cannot be maintained. Facts about the utterance situation, and the beliefs and preferences of the interlocutors influence the relevance and meaning of communicative acts in much the same way as syntactic and semantic rules do. This calls for an integrated approach to the semantics and pragmatics of natural language. The theoretical problem that will be addressed is to develop a mathematically precise framework which can cope with both the abstract study of meaning and structural aspects of language use. Our goal is to develop a quantitative theory of interpretation of natural language as a generalization and refinement of current qualitative perspectives, and to use it to determine the relevance and interpretation of a wide range of communicative acts. Our hypothesis is that a game-theoric approach offers the required unifying perspective. Our method will be to start from the analysis of empirical phenomena that resist a successful analysis in terms of quantitative rules, and to develop the formal framework in a number of stages with a growing complexity and empirical coverage. In doing so, we will extend several recently developed theories which transcend the traditional borderline between semantics and pragmatics, and help to build a unifying perspective on these research trends by employing the well-studied mathematical apparatus of game theory.