  We study moduli of semistable twisted sheaves on smooth proper morphisms of
algebraic spaces. In the case of a relative curve or surface, we prove results
on the structure of these spaces. For curves, they are essentially isomorphic
to spaces of semistable vector bundles. In the case of surfaces, we show (under
a mild hypothesis on the twisting class) that the spaces are asympotically
geometrically irreducible, normal, generically smooth, and l.c.i. over the
base. We also develop general tools necessary for these results: the theory of
associated points and purity of sheaves on Artin stacks, twisted Bogomolov
inequalities, semistability and boundedness results, and basic results on
twisted Quot-schemes on a surface.
