  These notes are based on lectures on index theory, topology, and operator
algebras at the "School on High Dimensional Manifold Theory" at the ICTP in
Trieste, and at the Seminari di Geometria 2002 in Bologna. We describe how
techniques coming from the theory of operator algebras, in particular
$C^*$-algebras, can be used to study manifolds. Operator algebras are
extensively studied in their own right. We will focus on the basic definitions
and properties, and on their relevance to the geometry and topology of
manifolds. The link between topology and analysis is provided by index
theorems. Starting with the classical Atiyah-Singer index theorem, we will
explain several index theorems in detail.
  Our point of view will be in particular, that an index lives in a canonical
way in the K-theory of a certain $C^*$-algebra. The geometrical context will
determine, which $C^*$-algebra to use. A central pillar of work in the theory
of $C^*$-algebras is the Baum-Connes conjecture. Nevertheless, it has important
direct applications to the topology of manifolds, it impliese.g. the Novikov
conjecture. We will explain the Baum-Connes conjecture and put it into our
context.
  Several people contributed to these notes by reading preliminary parts and
suggesting improvements, in particular Marc Johnson, Roman Sauer, Marco Varisco
und Guido Mislin. I am very indebted to all of them. This is an elaboration of
the first chapter of the author's contribution to the proceedings of the above
mentioned "School on High Dimensional Manifold Theory" 2001 at the ICTP in
Trieste.
