Positive Markov Jump Linear Systems are piecewise positive linear systems affected by a stochastic signal generated by a Markov chain. Positive systems naturally arise in the description of biological systems, compartmental models, population dynamics, traffic modeling, chemical reactions, queue processes, and so on. A rich literature on positive linear systems is now available. This is the first work to provide an overview of these developments. It outlines the typical applications of such systems, giving a detailed description of the mathematical theory underpinning the subject. It provides a comprehensive and timely introduction to the study of such systems. Readers who are new to the topic will find everything required to understand such systems in a concise and accessible form.