Progress in understanding the Brauer group has often come in the form of a deeper understanding of its geometric properties. We will sketch the historical development of the Brauer group and describe some of its hidden ties to geometry, culminating in a recent result of de Jong on Brauer groups of surfaces and a related result for surfaces over finite fields. The methods we describe replace de Jong's original proof with ideas related to his Porter lectures. In the finite field case, different (but closely related) methods are required. This talk is intended for a general audience.