Students and teachers of mathematics and related fields will find this book a comprehensive and modern approach to probability theory, providing the background and techniques to go from the beginning graduate level to the point of specialization in research areas of current interest. The book is designed for a two- or three-semester course, assuming only courses in undergraduate real analysis or rigorous advanced calculus, and some elementary linear algebra. A variety of applications Bayesian statistics, financial mathematics, information theory, tomography, and signal processing appear as threads to both enhance the understanding of the relevant mathematics and motivate students whose main interests are outside of pure areas.