In this talk we discuss various approaches to the long-stated toral rank conjecture. This conjecture gives lower bound on the cohomology rank of a finite dimensional topological space X with (almost) free torus (S^1)^m action. The question turns out to be closely related to the famous algebraic Buchsbaum-Eisenbud-Horrocks conjecture. We formulated graded (in fact filtered) version of toral rank conjecture and demonstrate how Lerray-Serre spectral sequences allows to derive some lower bounds.