The Atiyah-Singer index theorem shows that the topological and analytic indices of elliptic operators are equal. This theorem is a grand unification of three classical theorems: Gauss-Bonnet-Chern theorem, Hirzebruch signature theorem and Riemann-Roch-Hirzebruch theorem. We will introduce the heat kernel proof of the local index theorem. We also talk about the Ray-Singer analytic torsion, which is an important spectral invariant associated to Hodge-Laplacian.