During the spring 2009, an IHP trimester was organized on “nonlinear waves and dispersion”. At the time, 15 years of progress on the Cauchy problem for dispersive models were reaching their pinnacle, and at the same time it was an excellent occasion to learn about more recent progress on blow-up or longtime behavior, concentration-compactness tools, etc. The 2016 IHES program is intended to cover the spectacular developments of the later years, with a great deal of activity on more sophisticated models as well as physically realistic problems : refined description of asymptotic regimes, blow-up mechanisms, progress toward the soliton resolution conjecture, stability of breathers, understanding dispersive phenomena in general relativity, to name a few. Tools which were developed with dispersive toy models have been proved to be quite successful “in real life”: from the orbital stability of galaxies to the water waves, understanding and classifying solutions and their asymptotic is no longer an unreachable goal. Interactions with General Relativity, fluid mechanics or kinetic theory have proved to be fruitful, while existing connections with dynamical systems and harmonic analysis have been infused with renewed interest. The IHES program aims at bringing together a large spectrum of researchers from these closely related fields and foster the ever growing collaborations and exchanges of ideas, while providing to the younger part of its participants a large view of the most up to date methods, results, and remaining challenges.