The index theorem is an important theorem of the 20th century due to the work of Bott, Atiyah, Hirzebruch, Singer, Patodi, and others which states the (global) index of an elliptic differential operator can be computed by integrating local topological data given by the principle symbol of the operator. I will explain the statement of the index theorem and explain the classical geometric examples of Hirzebruch-Riemann-Roch, Chern-Gauss-Bonnet, and the Hirzebruch signature theorem. I will also give applications to 4-manifold topology and Dirac operators. Speaker(s): John Kilgore (UM)