This book is primarily a research monograph that presents in a unified man ner some recent research on a class of hybrid dynamical systems (HDS). The book is intended both for researchers and advanced postgraduate stu dents working in the areas of control engineering, theoretical computer science, or applied mathematics and with an interest in the emerging field of hybrid dynamical systems. The book assumes competence in the basic mathematical techniques of modern control theory. The material presented in this book derives from a period of fruitful research collaboration between the authors that began in 1994 and is still ongoing. Some of the material contained herein has appeared as isolated results in journal papers and conference proceedings. This work presents this material in an integrated and coherent manner and also presents many new results. Much of the material arose from joint work with students and colleagues, and the authors wish to acknowledge the major contributions made by Ian Petersen, Efstratios Skafidas, Valery Ugrinovskii, David Cook, Iven Mareels, and Bill Moran. There is currently no precise definition of a hybrid dynamical system; however, in broad terms it is a dynamical system that involves a mixture of discrete-valued and continuous-valued variables. Since the early 1990s, a bewildering array of results have appeared under the umbrella of HDS, ranging from the analysis of elementary on-off control systems to sophis ticated mathematical logic-based descriptions of large real-time software systems.