I am going to talk about Milnor's paper in which he proves that there are 7-manifolds ("exotic 7-spheres") which are homeomorphic but not diffeomorphic to the standard 7-sphere. These are total spaces of certain 3-sphere-bundles over the 4-sphere. Morse theory is then used to show that they are homeomorphic to the 7-sphere. In order to show that these manifolds are not diffeomorphic to the standard 7-sphere an invariant (\lambda) is built from the index and pontryagin classes of an 8-manifold bounding the 7-manifold.