ABSTRACT
From the interplay of topological K-theory and the analytic theory of pseudodifferential operators the Atiyah-Singer index theorem arose as a highlight in modern analysis. According to its very nature of bringing together two mathematical disciplines the index theorem can be looked at from both sides. On the one hand it gives topological criteria for the solvability of partial differential equations on manifolds and on the other hand integrality or vanishing of characteristic numbers in topology is guaranteed by interpreting them as indices of appropriate differential operators. This chapter presents a brief summary of the literature grouped into various categories which are, of course, a bit arbitrary in some respects. In the simplest and first example of an index formula, established in 1921 by Fritz Noether, the index of a one-dimensional singular integral operator is expressed by a winding number. This formula easily extends to systems. A related formula for Toeplitz operators is given in Gohberg-Krein 2.
