Lévy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. David Applebaum connects the two subjects together in this monograph. After an introduction to the general theory of Lévy processes, he accessibly develops the stochastic calculus for Lévy processes. All the tools needed for the stochastic approach to option pricing, including Itô's formula, Girsanov's theorem and the martingale representation theorem, are described.