  We extend knot Floer homology to string links in D^{2} \times I and to
d-based links in arbitrary three manifolds, without any hypothesis on the
null-homology of the components. As for knot Floer homology we obtain a
description of the Euler characteristic of the resulting homology groups (in
D^{2} \times I) in terms of the torsion of the string link. Additionally, a
state summation approach is described using the equivalent of Kauffman states.
Furthermore, we examine the situtation for braids, prove that for alternating
string links the Euler characteristic determines the homology, and develop
similar composition formulas and long exact sequences as in knot Floer
homology.
