  In this paper, we study Azumaya algebras and Brauer groups in derived
algebraic geometry. We establish various fundamental facts about Brauer groups
in this setting, and we provide a computational tool, which we use to compute
the Brauer group in several examples. In particular, we show that the Brauer
group of the sphere spectrum vanishes, and we use this to prove two uniqueness
theorems for the stable homotopy category. Our key technical results include
the local geometricity, in the sense of Artin n-stacks, of the moduli space of
perfect modules over a smooth and proper algebra, the \'etale local triviality
of Azumaya algebras over connective derived schemes, and a local to global
principle for the algebraicity of stacks of stable categories.
