This book collects some of the authors previous results on Markov jump systems (MJSs) subject to complex transition probabilities (TPs) as well as the original developments on the stability analysis of MJSs. This book can be used in undergraduate and graduate study. It can be also suitable as a reference for engineers and researchers in the field of systems with jump Markov perturbations. (Vasile Drgan, Mathematical Reviews, January, 2017)