This book presents a rigorous exposition of probability theory for a variety of applications. The first part of the book is a self-contained account of the fundamentals. Material suitable for advanced study is then developed from the basic concepts. Emphasis is placed on examples, sound interpretation of results and scope for applications.A distinctive feature of the book is that it discusses modern applications seldom covered in traditional texts. Two cases in point are risk theory (or comparison of distributions) and stochastic optimization. The book also includes some recent developments of probability theory, for example limit theorems for sums of dependent variables, nonlinear and nonclassical limit theorems. Simplified proofs and a unified approach to the exposition of many results are other key features.The book may be used as a textbook for graduate students and advanced undergraduates, and as a work of reference.