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This text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The Atiyah-Singer index theorem and its applications are developed (without complete proofs) via the heat equation method. Zeta functions for Laplacians and analytic torsion are also treated, and the recently uncovered relation between index theory and analytic torsion is laid out. The text is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning. There are over 100 exercises with hints.