Given an exact symplectic manifold M (with some additional data) we'll construct a category whose objects are Lagrangians (just like the Fukaya category) but whose morphisms are given by Lagrangian cobordisms. We'll then construct a functor from this category of Lagrangian cobordisms to the Fukaya category of M. This functor has a lot of nice properties. For instance, if two Lagrangians are related by a compact Lagrangian cobordism, then this functor tells us that they must be equivalent objects in the Fukaya category of M--in other worse, cobordant Lagrangians have equivalent Floer theories. The functor also shows that certain homotopy groups of cobordisms have homomorphisms to Floer cohomology groups of Lagrangians.