Nonlinear partial dierential equations provide powerful tools to model wave propagation in various physical contexts, such as nonlinear optics, interface dynamics and plasma physics. A common characteristics of these problems is that they have the form of a Hamiltonian system with innitely many degrees of freedom. Two main scenarios are central to nonlinear dynamics: on one hand, the formation of coherent structures often called solitons or solitary waves, and on the other hand the possible occurrence of singularity, a process in which a wave eld becomes innite in a nite time. This talk will explore these properties for one of the key equations of mathematical physics, the nonlinear Schrodinger equation, which provides a canonical description of wave propagation in weakly nonlinear dispersive media.