The general idea advocated in this book is to start from complicated mathematical models describing wave motion as it results from the rules of continuum mechanics and then to find a simpler viewpoint that still keeps everything essential preserved. Special attention is paid to the description of the sources of nonlinearities. The complexities in modelling are demonstrated by several examples including solitons, propagating instabilities and waves in waveguides. The selected case studies show some unconventional approaches in order to explain the richness of nonlinear wave motion. The final chapters are of more general character, including the essays on nonlinearity, beauty, and complexity. In this way, the thread of the analysis is the following: simple basic arguments result in a complicated theory that, in turn, needs certain simplifications in order to grasp the physical phenomena involved. What makes this book special compared to other books in the field is that all the examples are cast into a general philosophical network of complexity and simplicity. Audience: This volume will be of interest to researchers and students whose work involves mathematical modelling of wave phenomena.