  We use twisted sheaves and their moduli spaces to study the Brauer group of a
scheme. In particular, we (1) show how twisted methods can be efficiently used
to re-prove the basic facts about the Brauer group and cohomological Brauer
group (including Gabber's theorem that they coincide for a separated union of
two affine schemes), (2) give a new proof of de Jong's period-index theorem for
surfaces over algebraically closed fields, and (3) prove an analogous result
for surfaces over finite fields. We also include a reduction of all
period-index problems for Brauer groups of function fields over algebraically
closed fields to characteristic 0, which (among other things) extends de Jong's
result to include classes of period divisible by the characteristic of the base
field. Finally, we use the theory developed here to give counterexamples to a
standard type of local-to-global conjecture for geometrically rational
varieties over the function field of the projective plane.
