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With a locally compact group, one can naturally associate two objects encoding important parts of its representation theory: the full and the reduced group C*-algebra. These algebras coincide if and only if the group is amenable. In general, there are many C*-algebras (called exotic group C*-algebras) which lie "between these two", and in recent years, such algebras have received an increased amount of attention, partly because of their relation with the Baum-Connes conjecture.
I will give an introduction to this topic (without assuming any specific background in C*-algebras) and explain how one can, in many cases, construct exotic group C*-algebras from the unitary representation theory of a group.