It is remarkable that integrable PDEs play a basic role in the analytic description of extreme nonlinear wave phenomena in Nature. Indeed: i) a novel IST for vector fields allows one to study the wave breaking in the plane for the $n+1$-dimensional dispersionless KP equation, describing quasi one-dimensional nonlinear waves in the absence of dispersion and dissipation (joint work with Manakov); ii) the finite gap method and matched asymptotic expansions allow one to study how initial periodic perturbations of an unstable background generate an exact rogue wave recurrence for the focusing nonlinear Schrödinger equation, describing the modulation instability of quasi monochromatic waves in weakly nonlinear media (joint work with Grinevich).