  Boutet de Monvel's calculus provides a pseudodifferential framework which
encompasses the classical differential boundary value problems. In an extension
of the concept of Lopatinski and Shapiro, it associates to each operator two
symbols: a pseudodifferential principal symbol, which is a bundle homomorphism,
and an operator-valued boundary symbol. Ellipticity requires the invertibility
of both. If the underlying manifold is compact, elliptic elements define
Fredholm operators. Boutet de Monvel showed how then the index can be computed
in topological terms. The crucial observation is that elliptic operators can be
mapped to compactly supported $K$-theory classes on the cotangent bundle over
the interior of the manifold. The Atiyah-Singer topological index map, applied
to this class, then furnishes the index of the operator. Based on this result,
Fedosov, Rempel-Schulze and Grubb have given index formulas in terms of the
symbols. In this paper we survey previous work how C*-algebra K-theory can be
used to give a proof of Boutet de Monvel's index theorem for boundary value
problems, and how the same techniques yield an index theorem for families of
Boutet de Monvel operators. The key ingredient of our approach is a precise
description of the K-theory of the kernel and of the image of the boundary
symbol.
