Dispersive wave equations govern the behavior of many fundamental processes of nature:Quantum Mechanics, Electromagnetics, General Relativity, Optics and more. Nonlinear equations emerge naturally as either basic laws of nature as in General Relativity, or, as a limiting approximation of very complex systems. Such systems are for example nonlinear optics of devices and materials, water waves, Infinite particle Quantum Mechanical systems and more. The analysis of these equations has seen a dramatic growth in the last twenty years; I will describe some of the challenges and some of my work in this field, on Soliton dynamics, $L^p$ Decay estimates and Scattering theory.