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"Anomalies are quantum violations of classical conservation laws in presence of background Electromagnetic, or Gravitational fields. In studying anomalies, we encounter asymptotic expansions of heat kernels for Dirac operators, and N=1 supersymmetric quantum mechanics turns out to be a powerful physicist's tool for computing these. Interestingly, it also provides a simple way to arrive at the Atiyah-Singer index density for the Dirac operator. In this talk, we will review some of the details about how this works for Dirac operators on curved manifolds. If time permits, we will discuss generalizations to include torsion."