This is a graduate course in probability theory designed for Master's level students in statistics, mathematics, and other departments who are interested in a rigorous mathematical treatment of probability theory. Specificc topics to be covered include: Fields, sigma algebras, random variables, probabilities and distribution functions, expectation. Inequalities those of Jensen, Holder(Schwartz), and Minkowski. Convergence of sequences of random variables, uniform integrability. Product spaces and Fubini's theorem. Independence, Borel-Canteli lemma, characteristic functions, central limit theorem, inversion formula. Sums of independent of random variables; Strong and Weak Law of large numbers and the three series Theorem. If time permits: conditional probability, conditional expectation, martingales and martingale convergence theorems.