  Given a graph $G=(V(G), E(G))$ and a set $P\subseteq V(G)$, the following
concepts have been recently introduced: $(i)$ two elements of $P$ are
\emph{mutually visible} if there is a shortest path between them without
further elements of $P$; $(ii)$ $P$ is a \emph{mutual-visibility set} if its
elements are pairwise mutually visible; $(iii)$ the \emph{mutual-visibility
number} of $G$ is the size of any largest mutual-visibility set. % In this work
we continue to investigate about these concepts. We first focus on
mutual-visibility in Cartesian products. For this purpose, too, we introduce
and investigate independent mutual-visibility sets. In the very special case of
the Cartesian product of two complete graphs the problem is shown to be
equivalent to the well-known Zarenkiewicz's problem. We also characterize the
triangle-free graphs with the mutual-visibility number equal to $3$.
