Teaser: The urge to understand wave phenomena in natural systems is without much exaggeration the origin of an overwhelming part of applied mathematics. Predominant are fluids, but wave phenomena lie at the core of a much broader range of systems, including the heart, the brain, pandemic spreads and the climate. Generations of mathematicians have derived and studied an astounding number of model equations with regards to waves — in particular nonlinear ones. In the wake of organising the SIAM conference on nonlinear waves and coherent structures in Bremen, in this talk I attempt to provide some subjective insights into this field and touch upon some mathematical techniques that have been developed more recently.