Knot Floer homology is an invariant for knots and links defined by Ozsvath and Szabo and independently by Rasmussen. It has proven to be a powerful invariant e.g. in computing the genus of a knot, or determining whether a knot is fibered. In this talk I define a generalisation of knot Floer homology for tangles ; Tangle Floer homology is an invariant of tangles in $D^3$, $S^2\times I$ or in $S^3$. Tangle Floer homology satisfies a gluing theorem and its version in $S^3$ gives back a stabilisation of knot Floer homology.

In the first part of my talk I will give description of (a combinatorial version of) knot Floer homology, and a show its naive restriction to tangles. Then in the second part I give the correct definition of tangle Floer homology and finally I discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant.