  Proper proximality of a countable group is a notion that was introduced by
Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain
von Neumann algebras associated to groups or ergodic group actions. In the
present paper, we establish the proper proximality of many groups acting on
nonpositively curved spaces.
  First, these include many countable groups $G$ acting properly
nonelementarily by isometries on a proper $\mathrm{CAT}(0)$ space $X$. More
precisely, proper proximality holds in the presence of rank one isometries or
when $X$ is a locally thick affine building with a minimal $G$-action. As a
consequence of Rank Rigidity, we derive the proper proximality of all countable
nonelementary $\mathrm{CAT}(0)$ cubical groups, and of all countable groups
acting properly cocompactly nonelementarily by isometries on either a Hadamard
manifold with no Euclidean factor, or on a $2$-dimensional piecewise Euclidean
$\mathrm{CAT}(0)$ simplicial complex.
  Second, we establish the proper proximality of many hierarchically hyperbolic
groups. These include the mapping class groups of connected orientable
finite-type boundaryless surfaces (apart from a few low-complexity cases), thus
answering a question raised by Boutonnet, Ioana and Peterson. We also prove the
proper proximality of all subgroups acting nonelementarily on the curve graph.
  In view of work of Boutonnet, Ioana and Peterson, our results have
applications to structural and rigidity results for von Neumann algebras
associated to all the above groups and their ergodic actions.
