The study of complex adaptive systems is among the key modern tasks in science. Such systems show radically different behaviours at different scales and in different environments, and mathematical modelling of such emergent behaviour is very difficult, even at the conceptual level. We require a new methodology to study and understand complex, emergent macroscopic phenomena. Coarse graining, a technique that originated in statistical physics, involves taking a system with many microscopic degrees of freedom and finding an appropriate subset of collective variables that offer a compact, computationally feasible description of the system, in terms of which the dynamics looks "natural". The authors explain the basics of natural and artificial evolutionary dynamics, and offer detailed treatments of the related models of search spaces, population spaces, state spaces, crossover, mutation and selection. The rest of the book is concerned with the mathematical modelling of these aspects of evolutionary dynamics using the coarse graining technique, and with analysis of the subsequent models. This book is a significant contribution to the theory of artificial evolutionary systems, and will be key reading for theoreticians in computer science, artificial intelligence and engineering. While the insights into how complexity can be tamed will be valuable reading for biologists and physicists engaged with the theory of natural evolutionary systems.