This textbook is a contemporary, comprehensive introduction to categorial grammars in the logical tradition initiated by Lambek. It guides students and researchers through fundamental results, original developments and applications. Numerous examples and exercises underline the logical, computational and linguistic relevance of categorial grammars. Chapters: 1) AB grammars, 2) Lambek's Syntactic Calculus, 3) Lambek Calculus and Montague Grammar, 4) The non-associative Lambek calculus, 5) The multimodal Lambek calculus, 6) Proof nets for linear logic and the Lambek calculus, 7) Multimodal proof nets.