This talk is about homogeneous manifolds M equipped with a pseudo-Riemannian (i.e. indefinite) metric tensor, such that M = G/H for a Lie group G which acts transitively and isometrically on M, where H is a cocompact subgroup of G. We will see how density properties of H restrict the structure of G and the metric on M. A first special case are those M with solvable group G. Here, it turns out that H is a lattice in G and M is a locally symmetric space. This leads to the case groups G of arbitrary types, which is complicated by the existence of compact semisimple factors. I will also briefly address the existence of Einstein metrics in this class of manifolds. This talk is based on joint works with Oliver Baues, Yuri Nikolayevsky and Abdelghani Zeghib.