This book reports the recent progress in the field of continuous-time Markov chains (CTMCs), with a comprehensive and extensive discussion on its general theory and applications. It covers the properties of Q-matrix and transition function as well as the existence and uniqueness of Q-processes in stable and unstable cases. Monotonity, recurrency and ergodicity criteria are presented in this book. It also gives some important conclusions regarding the transiency properties, including invariant measures and quasi-limiting distributions for some important Q-processes.