In this chapter the foundations of categorial grammar are discussed. Firstly, the core notions stemming from Kazimierz Ajdukiewicz 's seminal works: functoriality principle, semantic category, syntactic position – are introduced and analyzed. Secondly, the postulates for the theory are identified and confronted with some counter-intuitions. Thirdly, some further modifications of classical categorial grammar are examined and evaluated. Finally, the main features of categorial grammar in general are highlighted and formed into arguments against different kinds of grammars.