  A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an
$n$-dimensional) smooth closed manifold with an effective locally standard
action of $(S^1)^n$ (resp. $(\mathbb Z_2)^n$) whose orbit space is
combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we
study them when $P$ is a product of simplices. A generalized Bott tower over
$\F$, where $\F=\C$ or $\R$, is a sequence of projective bundles of the Whitney
sum of $\F$-line bundles starting with a point. Each stage of the tower over
$\F$, which we call a generalized Bott manifold, provides an example of
quasitoric manifolds (when $\F=\C$) and small covers (when $\F=\R$) over a
product of simplices. It turns out that every small cover over a product of
simplices is equivalent (in the sense of Davis and Januszkiewicz \cite{DJ}) to
a generalized Bott manifold. But this is not the case for quasitoric manifolds
and we show that a quasitoric manifold over a product of simplices is
equivalent to a generalized Bott manifold if and only if it admits an almost
complex structure left invariant under the action. Finally, we show that a
quasitoric manifold $M$ over a product of simplices is homeomorphic to a
generalized Bott manifold if $M$ has the same cohomology ring as a product of
complex projective spaces with $\Q$ coefficients.
