  For a simple $n$-polytope $P$, a quasitoric manifold over $P$ is a
$2n$-dimensional smooth manifold with a locally standard action of the
$n$-dimensional torus for which the orbit space is identified with $P$. This
paper shows the topological classification of quasitoric manifolds over the
dual cyclic polytope $C^n(m)^*$, when $n>3$ or $m-n=3$. Besides, we classify
small covers, the "real version" of quasitoric manifolds, over all dual cyclic
polytopes.
