In view of the diversity of its applications, there is a wide range in the backgrounds of those who are to apply the theory of large deviations. This book provides an exposition geared towards such different audiences. The presentation is rigorous, and progresses from a finite dimensional analysis that requires little more than basic calculus and convex analysis to more abstract settings, requiring a solid background in analysis and probability. A plethora of applications, both in the simple as well as more abstract setup, illustrates the power of the techniques introduced. This book has been used as a textbook for applications-oriented courses in engineering/statistics/operations research, emphasizing the first half of the book, as well as for graduate courses in probability theory, emphasizing its second half.