  We generalize the notion of moment-angle manifold over a simple convex
polytope to an arbitrary nice manifold with corners. For a nice manifold with
corners Q, we first compute the stable decomposition of the moment-angle
manifold Z_Q via a construction called rim-cubicalization of Q. From this, we
derive a formula to compute the integral cohomology group of Z_Q via the strata
of Q. This generalizes the Hochster's formula for the moment-angle manifold
over a simple convex polytope. Moreover, we obtain a description of the
integral cohomology ring of Z_Q using the idea of partial diagonal maps. In
addition, we define the notion of polyhedral product of a sequence of based
CW-complexes over Q and obtain similar results for these spaces as we do for
Z_Q. Using this general construction, we can compute the equivariant cohomology
ring of Z_Q with respect to its canonical torus action from the
Davis-Januszkiewicz space of Q. The result leads to the definition of a new
notion called the topological face ring of Q, which generalizes the notion of
face ring of a simple polytope. Meanwhile, we obtain some parallel results for
the real moment-angle manifold RZ_Q over Q.
