  In this talk, we review the heat kernel approach to the Atiyah-Singer index
theorem for Dirac operators on closed manifolds, as well as the
Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with
boundary. We also discuss the odd dimensional counterparts of the above
results. In particular, we describe a joint result with Xianzhe Dai on an index
theorem for Toeplitz operators on odd dimensional manifolds with boundary.
