This chapter introduces the fundamental concepts of Lévy processes, which are essential for understanding Lévy option pricing models. The chapter builds on the work of French mathematician Paul Lévy, who combined Brownian motion and the compound Poisson process to develop the Lévy-Khintchine formula, encompassing a wide range of continuous and discontinuous stochastic processes. The construction of Lévy processes involves combining a Brownian motion and a compensated compound Poisson process, with the characteristic function of the resulting process given by the de Finetti characteristic function. The chapter also discusses the extension of this construction by adding additional components, focusing on the jump part and the introduction of the Lévy measure. The key question is under what conditions the integral defining the Lévy process is well-defined, with Lévy providing sufficient conditions for the existence of the integral.This chapter introduces the fundamental concepts of Lévy processes, which are essential for understanding Lévy option pricing models. The chapter builds on the work of French mathematician Paul Lévy, who combined Brownian motion and the compound Poisson process to develop the Lévy-Khintchine formula, encompassing a wide range of continuous and discontinuous stochastic processes. The construction of Lévy processes involves combining a Brownian motion and a compensated compound Poisson process, with the characteristic function of the resulting process given by the de Finetti characteristic function. The chapter also discusses the extension of this construction by adding additional components, focusing on the jump part and the introduction of the Lévy measure. The key question is under what conditions the integral defining the Lévy process is well-defined, with Lévy providing sufficient conditions for the existence of the integral.