  The paper surveys some new results and open problems connected with such
fundamental combinatorial concepts as polytopes, simplicial complexes, cubical
complexes, and subspace arrangements. Particular attention is paid to the case
of simplicial and cubical subdivisions of manifolds and, especially, spheres.
We describe important constructions which allow to study all these
combinatorial objects by means of methods of commutative and homological
algebra. The proposed approach to combinatorial problems relies on the theory
of moment-angle complexes, currently being developed by the authors. The theory
centres around the construction that assigns to each simplicial complex $K$
with $m$ vertices a $T^m$-space $\zk$ with a special bigraded cellular
decomposition. In the framework of this theory, the well-known non-singular
toric varieties arise as orbit spaces of maximally free actions of subtori on
moment-angle complexes corresponding to simplicial spheres. We express
different invariants of simplicial complexes and related
combinatorial-geometrical objects in terms of the bigraded cohomology rings of
the corresponding moment-angle complexes. Finally, we show that the new
relationships between combinatorics, geometry and topology result in solutions
to some well-known topological problems.
