  A topological version of Levinson's theorem is presented. Its proof relies on
a C*-algebraic framework which is introduced in detail. Various scattering
systems are considered in this framework, and more coherent explanations for
the corrections due to thresholds effects or for the regularization procedure
are provided. Potential scattering, point interactions, Friedrichs model and
Aharonov-Bohm operators are part of the examples which are presented. Every
concepts from scattering theory or from K-theory are introduced from scratch.
