  Let $R$ be a commutative ring. An Azumaya coring consists of a couple
$(S,\Cc)$, with $S$ a faithfully flat commutative $R$-algebra, and an
$S$-coring $\Cc$ satisfying certain properties. If $S$ is faithfully
projective, then the dual of $\Cc$ is an Azumaya algebra. Equivalence classes
of Azumaya corings form an abelian group, called the Brauer group of Azumaya
corings. This group is canonically isomorphic to the second flat cohomology
group. We also give algebraic interpretations of the second Amitsur cohomology
group and the first Villamayor-Zelinsky cohomology group in terms of corings.
