A mathematical fullerene is a three dimensional convex simple polytope with all 2-faces being pentagons and hexagons. At the moment the problem of classification of fullerenes is well-known and is vital due to the applications in chemistry, physics, biology and nanotechnology. Thanks to the toric topology, we can assign to each fullerene P its moment-angle manifold Z_P and quasitoric manifold M^6(P). The cohomology rings H(Z_P) is a combinatorial invariant of the fullerene P. We shall focus upon results on the rings H(Z_P) and their applications based on fullerenes interpretation of cohomology classes, their products and Massey product. The multigrading in the ring H(Z_P), coming from theconstruction of Z_P, and the multigraded Poincare duality
play an important role here. The talks is based on joint works with Nikolay Erokhovets, Nigel Ray and Taras Panov.