We present the first sizable grammar written in the Tree Adjoining Grammar formalism (TAG)1. In particular we have used 'lexicalized' TAGs as described in [Schabes, Abeillé and Joshi 1988]. We present the linguistic coverage of our grammar, and explain the linguistic reasons which lead us to choose the particular representations. We have shown that a wide range of linguistic phenomena can be handled within the TAG formalism with lexically specified structures only. We first state the basic structures needed for French, with a particular emphasis on TAG's extended domain of locality that enables us to state complex subcategorization phenomena in a natural way. We motivate the choice of the head for the different structures and we contrast the treatment of nominal arguments with that of sentential ones, which is particular to the TAG framework. We also give a detailed analysis of sentential complements, because it has lead us to introduce substitution into the formalism, and because TAG makes interesting predictions in these cases. We discuss the different linguistic phenomena corresponding to adjunction and to substitution respectively. We then move on to 'light verb' constructions, in which extraction freely occurs out of the predicative NP. They are handled in a TAG straightforwardly as opposed to the usual double analysis. We lastly give an overview of the treatment of adjuncts,and suggest a treatment of idioms which make them fall into the same representations as 'free' structures.