The tempered dual of a topological group is a classical example of noncommutative space in the sense of A. Connes. In this talk, we will discuss old and new connections between the study of the representation theory of real reductive groups and certain C*-algebras naturally attached with them. The main points will be the implementation of parabolic induction functors by Hilbert modules and the representation-theoretic approach to the Connes-Kasparov Isomorphism.