\"This book, which began as a seminar in 1985 at MIT, contains complete proofs of thelocal index theorem for Dirac operators using the heat kernel approach, together withits generalizations to equivariant Dirac operators and families of Dirac operators, aswell as background material on superconnections and equivariant differential forms.\"

\"Introduction 1 Background on Differential Geometry 1.1 Fibre Bundles and Connections 1.2 Riemannian Manifolds 1.3 Superspaces 1.4 Superconnections 1.5 Characteristic Classes 1.6 The Euler and Thorn Classes 2 Asymptotic Expansion of the Heat Kernel 2.1 Differential Operators 2.2 The Heat Kernel on Euclidean Space 2.3 Heat Kernels 2.4 Construction of the Heat Kernel 2.5 The Formal SoLution 2.6 The Trace of the Heat Kernel 2.7 Heat Kernels Depending on a Parameter 3 CLifford Modules and Dirae Operators 3.1 The Clifford Algebra 3.2 Spinors 3.3 Dirac Operators 3.4 Index of Dirac Operators 3.5 The Lichnerowicz Formula 3.6 Some Examples of Clifford Modules 4 Index Density of Dirac Operators 4.1 The Local Index Theorem 4.2 Mehler\'s Formula 4.3 Calculation of the Index Density 5The Exponential Map and the Index Density 5,1 Iacobian of the Exponential Map on Principal Bundles 5.2 The Heat Kernel of a Principal Bundle 5.3 Calculus with Grassmann and Clifford Variables 5.4 The Index of Dirac Operators 6 The Equivariant Index Theorem 6.1 The Equivariant Index of Dirac Operators 6.2 The Atiyah-Bott Fixed Point Formula 6.3 Asymptotic Expansion of the Equivariant Heat Kernel 6.4 The Local Equivariant Index Theorem 6.5 Geodesic Distance on a Principal Bundle 6.6 The heat kernel of an equivariant vector bundle 6.7 Proof of Proposition6.13 7 Equivariant Differential Forms 7.1 Equivariant Characteristic Classes 7.2 The Localization Formula 7.3 Bott\'s Formulas for Characteristic Numbers 7.4 Exact Stationary Phase Approximation 5 The Fourier Transform of Coadjoint Orbits 7.6 Equivariant Cohomology and Families 7.7 The Bott Class 8 The Kirillov Formula for the Equivariant Index 8.1 The Kirillov Formula 8.2 The Weyl and Kirillov Character Formulas 8.3 The Heat Kernel Proof of the Kirillov Formula 9 The Index Bundle 9.1 The Index Bundle in Finite Dimensions 9.2 The Index Bundle of a Family of Dirac Operators 9.3 The Chern Character of t\"