abstract: Stacks of stable sheaves on projective surfaces are known to have good regularity properties for large values of the discrete invariants: they are irreducible, normal, and lci. I will discuss similar results for stacks of stable twisted sheaves on certain orbisurfaces, with an emphasis on their applications to 1) a basic problem about Brauer groups of function fields and 2) a classical question on the Hasse principle for geometrically rational varieties over global fields.