Le?vy 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. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Le?vy processes, then leading on to develop the stochastic calculus for Le?vy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Le?vy processes to have finite moments; characterisation of Le?vy processes with finite variation; Kunita's estimates for moments of Le?vy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Le?vy processes; multiple Wiener-Le?vy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Le?vy-driven SDEs.