The Brauer group of a surface is an essential tool in the study of the arithmetic of surfaces over non-closed fields, but it can also bear on other geometric problems, such as the rationality of fourfolds (even over the complex numbers). Geometric realizations of Brauer classes on surfaces, as étale projective bundles, often play a key role in our understanding of such applications. In this talk, I will discuss joint work with Jack Petok and Anthony Várilly-Alvarado, in which we consider constructions of these étale projective bundles for Brauer classes on K3 surfaces. This builds on earlier results and predictions of Hassett and Tschinkel, as well as on recent results of van Geemen and Kaputska, who show that some (but not all) 2-torsion Brauer classes on K3 surfaces have realizations as the exceptional locus of a divisorial contraction on a hyperkahler fourfold.