Philosophers, logicians, linguists and even mathematicians have tried to decipher the mechanisms by which concepts are constructed from the meaning of words. One way to achieve this was the study of lexical meaning and its combinatorial properties. Our purpose is to explore the seminal ideas that have resulted in categorial grammars and their relationship with other grammatical models and actual theories of meaning, in a historical process that takes us from the notion of category to that of predication, and from this to the notion of function, then to functional categories and finally to the linguistic notion of argument structure.