Abstract: I will talk about the history of index theory, and its roots in

topology and analysis. On the one hand, there is the history that starts

with the Riemann-Roch theorem and leads, via Hirzebruch's generalization, to

the Atiyah-Singer index theorem for Dirac operators. On the other hand there

is an analytical tradition starting early in the 20th century with the work

of Fredholm on integral equations, and Fritz Noether's study of the Fredholm

index of Toeplitz operators. It was the study of Fredholm index problems by

analysts like Gelfand that inspired Atiyah and Singer to prove their general

formula for elliptic operators. Boutet de Monvel's index theorem for

Toeplitz operators related to pseudoconvex domains can be seen as a

culmination of this second historical stream. At the end of the talk I will

discuss recent results of P.Baum and myself, and where they fit in this

historical perspective.