  This article discusses the existence problem of a compact quotient of a
symmetric space by a properly discontinuous group with emphasis on the
non-Riemannian case.
  Discontinuous groups are not always abundant in a homogeneous space $G/H$ if
$H$ is non-compact. The first half of the article elucidates general machinery
to study discontinuous groups for $G/H$, followed by the most update and
complete list of symmetric spaces with/without compact quotients. In the second
half, as applications of general theory, we prove: (i) there exists a 15
dimensional compact pseudo-Riemannian manifold of signature $(7,8)$ with
constant curvature, (ii) there exists a compact quotient of the complex sphere
of dimension 1, 3 and 7, and (iii) there exists a compact quotient of the
tangential space form of signature $(p,q)$ if and only if $p$ is smaller than
the Hurwitz-Radon number of $q$.
