This is an introduction to the modern theory of probability and stochastic processes. Although the book is geared toward mathematical material, some attention is paid to applications. The first four chapters cover the basics of what would be required for a graduate course on probability theory: random variables, expectations, conditional expectations, independence, and the classical limit theorems. The final chapters introduce Martingales, Poisson Random Measures, Levy Processes and Brownian Motion. The book is self-contained, every concept is carefully defined, but the book is aimed at students with a high level of maturity. The author's clear writing is further enhanced by the inclusion of high quality exercises and examples.