  We completely classify non-spanning $3$-polytopes, by which we mean lattice
$3$-polytopes whose lattice points do not affinely span the lattice. We show
that, except for six small polytopes (all having between five and eight lattice
points), every non-spanning $3$-polytope $P$ has the following simple
description: $P\cap \mathbb{Z}^3$ consists of either (1) two lattice segments
lying in parallel and consecutive lattice planes or (2) a lattice segment
together with three or four extra lattice points placed in a very specific
manner. From this description we conclude that all the empty tetrahedra in a
non-spanning $3$-polytope $P$ have the same volume and they form a
triangulation of $P$, and we compute the $h^*$-vectors of all non-spanning
$3$-polytopes. We also show that all spanning $3$-polytopes contain a
unimodular tetrahedron, except for two particular $3$-polytopes with five
lattice points.
