  We develop a theory of twistor spaces for supersingular K3 surfaces,
extending the analogy between supersingular K3 surfaces and complex analytic K3
surfaces. Our twistor spaces are obtained as relative moduli spaces of twisted
sheaves on universal gerbes associated to the Brauer groups of supersingular K3
surfaces. In rank 0, this is a geometric incarnation of the Artin-Tate
isomorphism. Twistor spaces give rise to curves in moduli spaces of twisted
supersingular K3 surfaces, analogous to the analytic moduli space of marked K3
surfaces. We describe a theory of crystals for twisted supersingular K3
surfaces and a twisted period morphism from the moduli space of twisted
supersingular K3 surfaces to this space of crystals. As applications of this
theory, we give a new proof of the Ogus-Torelli theorem modeled on Verbitsky's
proof in the complex analytic setting and a new proof of the result of
Rudakov-Shafarevich that supersingular K3 surfaces have potentially good
reduction. These proofs work in characteristic 3, filling in the last remaining
gaps in the theory. As a further application, we show that each component of
the supersingular locus in each moduli space of polarized K3 surfaces is
unirational.
