In algebraic topology, we learn to associate groups Hn(T) to locally compact spaces which “count the n-dimensional holes in T". In this talk, I want to describe how to realize H3(T) as a set Br(T) of equivalence classes of certain well-behaved C* -algebras. The group structure imposed on Br(T) via its identification with H3(T) is very natural in its C* -setting. With this group structure, Br(T) is called the Brauer groupof T. Depending on your point of view, this result can be viewed either as a concrete realization of H3(T) or as a classification result for a class of C* -algebras. In the last part of the talk, I want to describe an equivariant version of Br(T) developed jointly with David Crocker, Alex Kumjian and Iain Raeburn. No prior knowledge of C* -algebras or operator algebras will be assumed.