  We initiate the study of 2.5D box visibility representations (2.5D-BR) where
vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and
edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We
prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The
complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every
graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in
linear time. We then turn our attention to 2.5D grid box representations
(2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit
square at integer coordinates. We show that an $n$-vertex graph that admits a
2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally,
we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given
footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of
bottom faces of the boxes in $\Gamma$.
