Abstract: As motivation, we consider the Pontryagin dual space for a discrete abelian group, that is a compact and Hausdorff space. When studying the commutative algebra of complex valued continuous functions on this space, one "nicely" encodes its topological properties. When the group is not longer abelian, its representation theory becomes more complicated and it turns out that the corresponding dual space is not necessarily Hausdorff. Hence, when searching for the corresponding algebraic tool in the non abelian case, one introduces the corresponding reduced group C^*-algebra. Our first aim is to give the definition of this object. Then, as central tool in "noncommutative geometry" used to study this object we present basics of K-theory for (group) C^*-algebras and of equivariant analytic K-homology.