The topological Brauer group of a (topological) manifold X is nothing more than the torsion subgroup of its third degree singular cohomology. Its name and significance comes from algebraic geometry: for example, it plays a fundamental role in rationality problems. In this talk we will sketch its basic property, usage, and discuss one method of computing it bringing together topology, group theory and homological algebra, which has seen recent usage. N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).