We will give a gentle survey, directed towards a general audience, to group C^*-algebras and their relations to questions and problems in geometry, topology, and index theory. We will start with finite dimensional complex vector spaces and finite dimensional complex representations of finite groups, where many important ideas can already be illustrated. Then we will pass to a more advanced part, where infinite groups are considered. We will introduce the Baum-Connes Conjecture and its applications. We will explain how it can be used for explicite calculations of the topological K-theory of group C^*-algebras, which in turn have applications to the theory of C^*-algebras and problems in geometry and topology.