  Let $k$ be a field that is finitely generated over the field of rational
numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely
irreducible smooth projective variety over $k$, let $Br(X)$ be the
cohomological Brauer-Grothendieck group of $X$ and $Br_0(X)$ the image of
$Br(k)$ in $Br(X)$. We write $Br_1(X)$ for the subgroup of elements in $Br(X)$
that become trivial after replacing $k$ by its algebraic closure.
  We prove that $Br(X)/Br_0(X)$ is finite if $X$ is a $K3$ surface. When $X$ is
(a principal homogeneous space of) an abelian variety over $k$ then we prove
that $Br(X)/Br_1(X)$ is finite.
