  In this paper, we introduce a sequence of invariants of a knot K in S^3: the
knot Floer homology groups of the preimage of K in the m-fold cyclic branched
cover over K. We exhibit the knot Floer homology in the m-fold branched cover
as the categorification of a multiple of the Turaev torsion in the case where
the m-fold branched cover is a rational homology sphere. In addition, when K is
a 2-bridge knot, we prove that the knot Floer homology of the lifted knot in a
particular Spin^c structure in the branched double cover matches the knot Floer
homology of the original knot K in S^3. We conclude with a calculation
involving two knots with identical knot Floer homology in S^3 for which the
knot Floer homology groups in the double branched cover differ as Z_2-graded
groups.
