  Examples suggest that there is a correspondence between L-spaces and
3-manifolds whose fundamental groups cannot be left-ordered. In this paper we
establish the equivalence of these conditions for several large classes of such
manifolds. In particular, we prove that they are equivalent for any closed,
connected, orientable, geometric 3-manifold that is non-hyperbolic, a family
which includes all closed, connected, orientable Seifert fibred spaces. We also
show that they are equivalent for the 2-fold branched covers of non-split
alternating links. To do this we prove that the fundamental group of the 2-fold
branched cover of an alternating link is left-orderable if and only if it is a
trivial link with two or more components. We also show that this places strong
restrictions on the representations of the fundamental group of an alternating
knot complement with values in Homeo_+(S^1).
