Sutured Heegaard Floer and embedded contact homologies are isomorphic

We prove the equivalence of the sutured versions of Heegaard Floer homology, monopole Floer homology, and embedded contact homology. As applications we show that the knot versions of Heegaard Floer homology and embedded contact homology are equivalent and that product sutured 3-manifolds are characterized by the fact that they carry an adapted Reeb vector field without periodic orbits.