  For a given graph $G$, the mutual-visibility problem asks for the largest set
of vertices $M \subseteq V(G)$ with the property that for any pair of vertices
$u,v \in M$ there exists a shortest $u,v$-path of $G$ that does not pass
through any other vertex in $M$.
  The mutual-visibility problem for Cartesian products of a cycle and a path,
as well as for Cartesian products of two cycles, is considered. Optimal
solutions are provided for the majority of Cartesian products of a cycle and a
path, while for the other family of graphs, the problem is completely solved.
