The classical theory of the Brauer group of a noetherian ring or scheme was developed in the 1960's by Auslander–Goldman and Grothendieck. In this talk, I will explain a complex-analytic analogue of the story, where the role of a noetherian ring is played by the (usually non-noetherian) ring of holomorphic functions on a Stein space. Joint work with Olivier Benoist.