  We show that several torsion free 3-manifold groups are not left-orderable.
Our examples are groups of cyclic branched covers of S^3 branched along links.
  The figure eight knot provides simple nontrivial examples. The groups arising
in these examples are known as Fibonacci groups which we show not to be
left-orderable. Many other examples of non-orderable groups are obtained by
taking 3-fold branched covers of S^3 branched along various hyperbolic 2-bridge
knots. The manifold obtained in such a way from the 5_2 knot is of special
interest as it is conjectured to be the hyperbolic 3-manifold with the smallest
volume.
