The Atiyah-Singer index theorem is one of the great theorems of 20-th century mathematics and the entry point to many areas of contemporary research. It is a theorem at the confluence of differential geometry, partial differential equations, and topology with applications in many other areas of mathematics. This Ph.D. minicourse of 8 hours will describe the heat equation proof of the index theorem for Dirac-type operators. It will also meet on May 25 and 27 and on June 3, always at 11, always in Aula F.