Taken from the introduction to A Tour of Bordered Floer Theory (by Lipshitz, Ozsvath, D. Thurston):
http://arxiv.org/pdf/1107.5621v1.pdf

Heegaard Floer homology, introduced in a series of papers of Zoltán Szabó and the second author, has become a useful tool in 3- and 4-dimensional topology. The Heegaard Floer invariants contain subtle topological information, allowing one to detect the genera of knots and homology classes; detect fiberedness for knots and 3-manifolds; bound the slice genus and unknotting number; prove tightness and obstruct Stein fillability of contact structures; and more. It has been useful for resolving a number of conjectures, particularly related to questions about Dehn surgery. It is either known or conjectured to be equivalent to several other gauge-theoretic or holomorphic curve invariants in low-dimensional topology, including monopole Floer homology, embedded contact homology, and the Lagrangian matching invariants of 3- and 4-manifolds. Heegaard Floer homology is known to relate to Khovanov homology, and more relations with Khovanov-Rozansky type homologies are conjectured.

(references provided in the paper for each statement above)