Certain natural subspaces of a product of CW complexes, called polyhedral
products, play an important role in a variety of different fields including: toric varieties, toric
manifolds/orbifolds, intersections of quadrics, homotopy theory, algebraic combinatorics
complements of subspace arrangements, robotics and group theory. The talk will survey
certain combinatorics based constructions on polyhedral products which allow for a
description of the cohomology rings of certain families of toric manifolds in a particularly
compact form. The new constructions will be related to  a generalization of the basic Davis-
Januszkiewicz construction of toric manifolds.  This report is based on joint work with Martin
Bendersky, Fred Cohen and Sam Gitler.