The introductory chapter of this book seeks to address fundamental questions concerning its subject matter. It is organised into several sections, each titled respectively: “Why semigroups of operators?”, “Why nonlinear”, “Why pointwise Lipschitzian?”, and “Why asymptotic?”. In response to these questions, we elucidate the natural occurrence of nonlinear, pointwise Lipschitzian mappings and explain how the common fixed points of semigroups of such mappings are related to the stationary points of dynamical systems generated by these semigroups. We observe that stationary points of a process inherently correspond to its long-term behaviour. From this standpoint, adopting an asymptotic perspective is quite rational, as our primary interest is centred on the evolution of the system over time.