Rogue and Shock Waves in Nonlinear Dispersive Media Rogue and shock waves: an introduction Nonlinearity and dispersion in hydrodynamics and optical waves Breathers in nonlinear dispersive media Generation of breathers in the laboratory Statistical theory of rogue waves Mathematical methods for finding rogue waves solutions Dispersive shock waves: theoretical aspects Dispersive shock waves in optics and in the ocean. This self contained set of lectures addresses a gap in the literature by providing a systematic link between the theoretical foundations of the subject matter and cutting edge applications in both geophysical fluid dynamics and nonlinear optics. Rogue and shock waves are phenomena that may occur in the propagation of waves in any nonlinear dispersive medium. Accordingly they have been observed in disparate settings – as ocean waves in nonlinear optics in Bose Einstein condensates and in plasmas. Rogue and dispersive shock waves are both characterized by the development of extremes: for the former the wave amplitude becomes unusually large while for the latter gradients reach extreme values. Both aspects strongly influence the statistical properties of the wave propagation and are thus considered together here in terms of their underlying theoretical treatment. This book offers a self contained graduate level text intended as both an introduction and reference guide for a new generation of scientists working on rogue and shock wave phenomena across a broad range of fields in applied physics and geophysics.