The Nonlinear Schrodinger equation is an important equation in hydrodynamical and optical problems, especially relating to modulational instability. But it cannot account for extra dispersive effects, and in these cases it is not totally valid. However, there exist special extensions of the equation which can account for infinitely many orders of dispersion, remain integrable, and for which we have exact solutions. In this seminar we discuss these doubly-periodic and higher order periodic (breather) solutions, and their relations to solitons and rogue wave formation.