M. W. Davis and T. Januszkiewicz [Duke Math. J. 62, No. 2, 417–451 (1991; Zbl 0733.52006)] introduced a construction which associates to a simplicial complex \(K\) a cellular complex \({\mathcal Z}_K\), called the moment-angle complex, which is a subcomplex of the \(n\)-fold product of the \(2\)-disc, \(n\) being the number of vertices in \(K\). Following work of Buchstaber, Panov and others relating the algebro-topological properties of \({\mathcal Z}_K\) to the combinatorial properties of \(K\), the paper under review shows how the rank of the homotopy groups of \({\mathcal Z}_K\) may be computed from information about \(K\). The authors also study Massey products in the cohomology ring of \({\mathcal Z}_K\), proving that certain such products are non-zero.