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 $\mathbb{F}$,
where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$, is a sequence
of projective bundles of the Whitney sum of $\mathbb{F}$-line
bundles starting with a point. Each stage of the tower over
$\mathbb{F}$, which we call a generalized Bott manifold, provides
an example of quasitoric manifolds (when $\mathbb{F}=\mathbb{C}$)
and small covers (when $\mathbb{F}=\mathbb{R}$) 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
[5]) 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 $\mathbb{Q}$
coefficients.