Rogue waves in deep water are investigated in the framework of the nonlinear Schroedinger (NLS) equation and the modified Dysthe (MD) equation. Phase modulation of higher order homoclinic solutions of the NLS equation can produce coalescence of unstable spatial modes, creating waves of maximal amplitude. Numerical simulations of the MD equation indicate that a chaotic regime increases the likelihood of rogue wave formation, and that enhanced focusing occurs due to chaotic phase evolution. In this talk, we will discuss the models, higher order homoclinic solutions of the NLS equation, and show that the formation of extreme waves in random oceanic sea states characterized by JONSWAP power spectra is well predicted by the proximity to homoclinic solutions of the NLS equation. Finally, we will discuss recent work on the relation between wave amplification and phase singularities.