  We study lattices acting on $\textrm{CAT}(0)$ spaces via their commensurated
subgroups. To do this we introduce the notions of a graph of lattices and a
complex of lattices giving graph and complex of group splittings of
$\textrm{CAT}(0)$ lattices. Using this framework we characterise irreducible
uniform $(\textrm{Isom}(\mathbb{E}^n)\times T)$-lattices by $C^\ast$-simplicity
and the failure of virtual fibring and biautomaticity. We construct
non-residually finite uniform lattices acting on arbitrary products of right
angled buildings and non-biautomatic lattices acting on the product of
$\mathbb{E}^n$ and a right-angled building. We investigate the residual
finiteness, $L^2$-cohomology, and $C^\ast$-simplicity of $\textrm{CAT}(0)$
lattices more generally. Along the way we prove that many right angled Artin
groups with rank $2$ centre are not quasi-isometrically rigid.
