Toric topology assigns to each simplicial complex a finite CW-complex, called the moment-angle complex, which comes with a natural torus action. Homotopy invariants of this space recover various homological invariants of Stanley-Reisner rings of interest in combinatorial commutative algebra. In this talk I will describe certain higher cohomology operations induced by the torus action on a moment-angle complex. Focusing on examples, I'll explain how these operations assemble into an explicit Hirsch-Brown model of the torus action and can be used to give a combinatorial description of the minimal free resolution of Stanley-Reisner rings. Time permitting, I will indicate the relevance of these operations to cohomological rigidity problems in toric topology. This talk is based on joint work with Benjamin Briggs.