The pseudo-hyperbolic space H^&ob;p,q&cb; is the model of pseudo-riemannian manifolds of signature (p,q) and constant negative sectional curvature. We would like to describe its compact quotients (by a properly discontinuous group of isometries), but a first non-trivial problem is whether such quotients exist. After reviewing classical obstructions to their existence (for p and q odd and for p ≤ q) and the few known constructions (for q=1,3 or 7), I will formulate a conjecture about the topology of those compact quotients and explain how this conjecture inspired new and powerful obstructions, obtained jointly with Fanny Kassel and Yosuke Morita.