We propose a program of study of asymptotics and singularity formation for a class of nonlinear dispersive partial differential equations appearing in various areas of physics. A canonical example is the focusing nonlinear Schrodinger equation which appears both in nonlinear optics and plasma physics as a universal model for the self trapping of nonlinear waves. This concentration phenomenon appears in various physical situations but is in general very poorly understood. A breakthrough was achieved recently by Merle and Raphael who could obtain the first complete description of a singularity formation in the so-called mass critical case for the Schrodinger equation. Our first objective is to study the singularity formation for a larger class of models of waves in plasmas including the supercritical nonlinear Schrodinger equation, the energy critical wave and the Zakharov systems. Numerical simulations provide evidence of existence of blow-up solutions for some of these models which possess prescribed blow-up rate and profile, but little understanding of the ongoing mechanisms and in particular of their observed stability. An actual mathematical understanding of these blow up mechanisms is a challenging program which aims first at constructing explicit examples which are mostly unknown, and second studying generic asymptotic dynamics by using the knowledge acquired in obtaining particular solutions. Moreover, recent works on kinetic models arising in astrophysics show the effectiveness of tools developed for nonlinear dispersive models, and various models from astrophysics should be amenable to this line of study and draw a natural continuation of these techniques towards the fluid world.