  This article studies the volume of compact quotients of reductive homogeneous
spaces. Let $G/H$ be a reductive homogeneous space and $\Gamma$ a discrete
subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We
prove that the volume of $\Gamma \backslash G/H$ is the integral, over a
certain homology class of $\Gamma$, of a $G$-invariant form on $G/K$ (where $K$
is a maximal compact subgroup of $G$).
  As a corollary, we obtain a large class of homogeneous spaces the compact
quotients of which have rational volume. For instance, compact quotients of
pseudo-Riemannian spaces of constant curvature $-1$ and odd dimension have
rational volume. This contrasts with the Riemannian case.
  We also derive a new obstruction to the existence of compact Clifford--Klein
forms for certain homogeneous spaces. In particular, we obtain that
$\mathrm{SO}(p,q+1)/\mathrm{SO}(p,q)$ does not admit compact quotients when $p$
is odd, and that $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(m,\mathbb{R})$ does not
admit compact quotients when $m$ is even.
