Résumé : This book is a contemporary and comprehensive introduction to categorial grammars in the logical tradition initiated by the work of Lambek. It guides students and researchers through the fundamental results in the field, providing modern proofs of many classic theorems, as well as original recent advances. Numerous examples and exercises illustrate the motivations and applications of these results from a linguistic, computational and logical point of view. The Lambek calculus and its variants, and the corresponding grammars, are at the heart of these lecture notes. A chapter is devoted to a key feature of these categorial grammars: their very elegant syntax-semantic interface. In addition, we adapt linear logic proof nets to these calculi since they provide efficient parsing algorithms as exemplified in the Grail parser. This book shows how categorial grammars weave together converging ideas from formal linguistics, typed lambda calculus, Montague semantics, proof theory and linear logic, thus yielding a coherent and formally elegant framework for natural language syntax and semantics.