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I will introduce the Brauer group of smooth complex varieties and describe the two different types of information it contains: one coming from the algebraic structure of the variety, and a purely topological contribution given by the torsion in third integral cohomology. I will then discuss joint work with F. Tufo, A. Frassineti, and M. Verni, in which we study the Brauer group of Enriques manifolds and its relation with their topology.