We study moduli of semistable twisted sheaves on smooth proper 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 asymptotically geometrically irreducible, normal, generically smooth, and locally complete intersections (l.c.i.'s) 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 $\mathrm{Quot}$-schemes on a surface