  Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be
spaces obtained as unions of certain product spaces indexed by the simplices of
an abstract simplicial complex. In this paper we give a very general homotopy
theoretic construction of polyhedral products over arbitrary pointed posets. We
show that under certain restrictions on the poset $\calp$, that include all
known cases, the cohomology of the resulting spaces can be computed as an
inverse limit over $\calp$ of the cohomology of the building blocks. This
motivates the definition of an analogous algebraic construction - the
polyhedral tensor product. We show that for a large family of posets, the
cohomology of the polyhedral product is given by the polyhedral tensor product.
We then restrict attention to polyhedral posets, a family of posets that
include face posets of simplicial complexes, and simplicial posets, as well as
many others. We define the Stanley-Reisner ring of a polyhedral poset and show
that, like in the classical cases, these rings occur as the cohomology of
certain polyhedral products over the poset in question. For any pointed poset
$\calp$ we construct a simplicial poset $s(\calp)$, and show that if $\calp$ is
a polyhedral poset then polyhedral products over $\calp$ coincide up to
homotopy with the corresponding polyhedral products over $s(\calp)$.
