  We study the structure of the commensurator of a virtually abelian subgroup
$H$ in $G$, where $G$ acts properly on a $\mathrm{CAT}(0)$ space $X$. When $X$
is a Hadamard manifold and $H$ is semisimple, we show that the commensurator of
$H$ coincides with the normalizer of a finite index subgroup of $H$. When $X$
is a $\mathrm{CAT}(0)$ cube complex or a thick Euclidean building and the
action of $G$ is cellular, we show that the commensurator of $H$ is an
ascending union of normalizers of finite index subgroups of $H$. We explore
several special cases where the results can be strengthened and we discuss a
few examples showing the necessity of various assumptions. Finally, we present
some applications to the constructions of classifying spaces with virtually
abelian stabilizers.
