  This article gives an up-to-date account of the theory of discrete group
actions on non-Riemannian homogeneous spaces.
  As an introduction of the motifs of this article, we begin by reviewing the
current knowledge of possible global forms of pseudo-Riemannian manifolds with
constant curvatures, and discuss what kind of problems we propose to pursue.
  For pseudo-Riemannian manifolds, isometric actions of discrete groups are not
always properly discontinuous. The fundamental problem is to understand when
discrete subgroups of Lie groups $G$ act properly discontinuously on
homogeneous spaces $G/H$ for non-compact $H$. For this, we introduce the
concepts from a group-theoretic perspective, including the `discontinuous dual'
of $G/H$ that recovers $H$ in a sense.
  We then summarize recent results giving criteria for the existence of
properly discontinuous subgroups, and the known results and conjectures on the
existence of cocompact ones. The final section discusses the deformation theory
and in particular rigidity results for cocompact properly discontinuous groups
for pseudo-Riemannian symmetric spaces.
