A recurring theme in algebraic geometry is that cohomology classes often have concrete geometric realizations. In this talk, I will discuss the many geometric avatars of classes in H^1_et(X, PGL_n), including Brauer–Severi varieties, Azumaya algebras, and PGL_n-torsors, as well as their relation to the Brauer group via twisted sheaves. Interpreting cohomology through these objects gives a powerful way to prove vanishing results for certain cohomology groups. A nice application of these ideas shows H^i_{et}(X, Z/n) is what we expect when X is a smooth projective curve.