L?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 L?vy processes, then leading on to develop the stochastic calculus for L?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 L?vy processes to have finite moments; characterization of L?vy processes with finite variation; Kunita's estimates for moments of L?vy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general L?vy processes; multiple Wiener-L?vy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for L?vy-driven SDEs.