This book provides a systematic introduction to the dynamical behaviors of discontinuous dynamical systems and system synchronization. The first part of the book presents the theory of flow switching in discontinuous dynamical systems as well as the switching dynamics in such discontinuous systems. The second part presents a theory for interaction discontinuities between distinct dynamical systems, from which partial and complete chaotic synchronizations in such systems are studied analytically and experimentally. Examines partial/complete chaotic synchronizations of 2 nonlinear gyro-scope systems in the presence/absence of noise; Analyzes the projection synchronization mechanism of Chua circuits with different scrolls; Details interactions of the controlled Duffing oscillator with Van der Pol system and Pendulum system.