This book provides an overview of the main approaches used to analyze the dynamics of cellular
automata. Cellular automata are an indispensable tool in mathematical modeling. In contrast to
classical modeling approaches like partial differential equations cellular automata are
relatively easy to simulate but difficult to analyze. In this book we present a review of
approaches and theories that allow the reader to understand the behavior of cellular automata
beyond simulations. The first part consists of an introduction to cellular automata on Cayley
graphs and their characterization via the fundamental Cutis-Hedlund-Lyndon theorems in the
context of various topological concepts (Cantor Besicovitch and Weyl topology). The second
part focuses on classification results: What classification follows from topological concepts
(Hurley classification) Lyapunov stability (Gilman classification) and the theory of formal
languages and grammars (Kurka classification)? These classifications suggest that cellular
automata be clustered similar to the classification of partial differential equations into
hyperbolic parabolic and elliptic equations. This part of the book culminates in the question
of whether the properties of cellular automata are decidable. Surjectivity and injectivity are
examined and the seminal Garden of Eden theorems are discussed. In turn the third part
focuses on the analysis of cellular automata that inherit distinct properties often based on
mathematical modeling of biological physical or chemical systems. Linearity is a concept that
allows us to define self-similar limit sets. Models for particle motion show how to bridge the
gap between cellular automata and partial differential equations (HPP model and ultradiscrete
limit). Pattern formation is related to linear cellular automata to the Bar-Yam model for the
Turing pattern and Greenberg-Hastings automata for excitable media. In addition models for
sand piles the dynamics of infectious d
