  Neighborly cubical polytopes exist: for any $n\ge d\ge 2r+2$, there is a
cubical convex d-polytope $C^n_d$ whose $r$-skeleton is combinatorially
equivalent to that of the $n$-dimensional cube. This solves a problem of
Babson, Billera & Chan.
  Kalai conjectured that the boundary $\partial C^n_d$ of a neighborly cubical
polytope $C^n_d$ maximizes the $f$-vector among all cubical $(d-1)$-spheres
with $2^n$ vertices. While we show that this is true for polytopal spheres for
$n\le d+1$, we also give a counter-example for $d=4$ and $n=6$.
  Further, the existence of neighborly cubical polytopes shows that the graph
of the $n$-dimensional cube, where $n\ge5$, is ``dimensionally ambiguous'' in
the sense of Gr\"unbaum. We also show that the graph of the 5-cube is
``strongly 4-ambiguous''.
  In the special case $d=4$, neighborly cubical polytopes have $f_3=f_0/4
\log_2 f_0/4$ vertices, so the facet-vertex ratio $f_3/f_0$ is not bounded;
this solves a problem of Kalai, Perles and Stanley studied by Jockusch.
