In this article we study free toposes with the help of intuitionist type theory. Our treatment is self-contained and aims to be accessible to both categorists and logicians. We attempt to explain the relevant logic to the former and the categorical applications to the latter. Algebraically, free toposes arise as solutions to a universal problem, which amounts to constructing a left adjoint to the forgetful functor Top+Graph. Here “Top” denotes the category of small toposes, which we shall assume to possess a natural number object, with appropriate morphisms. These are essentially the socalled logical functors, except that we insist on them being strict functors which preserve everything on the nose. “Graph” denotes the category of graphs, which we take to be oriented, and functor-like morphisms. The adjoint functor Graph-Top associates to each graph Ythe topos 7(.r’) freely generated by s: In particular, when .f‘ = 0 is the empty graph, we obtain the so-called free topos 7(O), which is an initial object in Top. Lawvere has often pointed out the strong connection between topos theory and higher order intuitionist logic. It is precisely in the construction of the free topos that this connection is seen most easily. In Section 1 we present a formulation of intuitionist type theory with product types and mention the fundamental theorem which comprises three things: (1) the consistency of intuitionist type theory, (2) the v-property which asserts that if pvq is provable then either p or q is provable, (3) The 3-property which asserts that if 3.,-E.q q(x) is provable, then q(a) is provable for some term a of type A. Our type-theoretical language contains enough terms to witness all existential theorems; yet it does not contain too many terms, for example, it lacks a description operator. The fundamental theorem can be proved by several methods: (a) the cut elimination method of Gentzen-Girard,