The nonlinear Schrödinger equation (NLSE) is a weakly nonlinear evolution equation, that describes the propagation dynamics of wave packets in nonlinear dispersive media. Recent laboratory experiments on localized breathers on zero and finite background confirmed the validity of the NLSE to describe extreme localizations in fiber optics as well as in water of finite and infinite depth, beyond theoretical limitations. We will discuss experimental results in several nonlinear dispersive media. In particular, we will emphasize particular analogies between nonlinear water and electromagnetic waves. Limitations of the approach will be underlined as well. Furthermore, recent theoretical, numerical and experimental results, taking into account corrections to the NLSE, will be presented too.