  Based on the work of Adams and Stuck as well as on the work of Zeghib, we
classify the Lie groups which can act isometrically and locally effectively on
Lorentzian manifolds of finite volume. In the case that the corresponding Lie
algebra contains a direct summand isomorphic to the two-dimensional special
linear algebra or to a twisted Heisenberg algebra, we also describe the
geometric structure of the manifolds if they are compact. Using these results,
we investigate the local geometry of compact homogeneous Lorentz spaces whose
isometry groups have non-compact connected components. It turns out that they
all are reductive. We investigate the isotropy representation, curvatures and
holonomy. Especially, we obtain that any Ricci-flat compact homogeneous Lorentz
space is flat or has compact isometry group.
