  These notes cover the contents of three survey lectures held at the ICTP
Trieste Summer school on High dimensional manifold theory 2001.
  They introduce techniques coming from the theory of operator algebras. We
will focus on the basic definitions and properties, and on their relevance to
the geometry and topology of manifolds. An central pillar of work in the theory
of C*-algebras is the Baum-Connes conjecture. It implies the Novikov
conjecture. In the first talk, the Baum-Connes conjecture will be explained and
put into our context. One application of the Baum-Connes conjecture is to the
positive scalar curvature question. It implies the so called ``stable
Gromov-Lawson-Rosenberg conjecture''. The unstable version of this conjecture
said that, given a closed spin manifolds M, a certain obstruction, living in a
certain (topological) K-theory group, vanishes if and only M admits a
Riemannian metric with positive scalar curvature. It turns out that this is
wrong, and counterexamples will be presented in the second talk. The third talk
introduces $L^2$-cohomology, $L^2$-Betti numbers and other $L^2$-invariants.
These invariants, their basic properties, and the central questions about them
are introduced.
