This book provides new research on the dynamics, algorithms and synchronization of chaotic systems. Chapter One introduces some nonlinear techniques for the synchronization of a class of chaotic oscillator under the framework of observer design from control theory. Chapter Two discusses in detail the application of a novel multi-domain spectral collocation approach for finding solutions of ordinary differential equations that exhibit general chaotic behaviour. Chapter Three designs an adaptive state feedback controller guaranteeing the asymptotic stability followed by the synchronization of the nonlinear discrete-time error of two identical hyper chaotic systems.