The nonlinear Schrodinger (NLS) equation provides a canonical description for the envelope dynamics of a quasi-monochromatic plane wave (the carrying wave) propagating in a weakly nonlinear dispersive medium when dissipative processes are negligible. On short times and small propagation distances, the dynamics arc linear, but cumulative nonlinear interactions result hi a significant modulation of the wave amplitude on large spatial and temporal scales. The NLS equation expresses how the linear dispersion relation is affected by the thickening of the spectral lines associated to the modulation and the resonant nonlinear interactions. In optics, it can also be viewed as the extension to nonlinear media of the paraxial approximation, extensively used for linear waves propagating in random media.