Lévy processes are a class of continuous-time stochastic processes that combine Brownian motion and compound Poisson processes. Paul Lévy introduced the concept, leading to the Lévy-Khintchine formula, which describes a wide range of continuous and discontinuous stochastic processes. This chapter provides the fundamentals of Lévy processes essential for understanding Lévy option pricing models. Key references include Lévy (1954), Feller (1971), Bertoin (1996), and Sato (1999). Notations follow those in Chapter 16, which is recommended to be read before this chapter. The original construction of Lévy processes was heuristic, combining Brownian motion and a compensated compound Poisson process. The process $ X_t $ is defined as the sum of a Brownian motion $ X_t^{BM} $ and a compensated compound Poisson process $ X_t^{CP} $. The characteristic function of $ X_t $ is derived by combining the characteristic functions of the Brownian motion and the compound Poisson process, resulting in the de Finetti characteristic function. The exponent in the characteristic function is crucial. When combining multiple jump components, the Lévy measure $ \nu(dz) $ is introduced, which measures the intensity and distribution of jumps. The Lévy measure must satisfy certain conditions to ensure the integral exists. Lévy provided sufficient conditions for the existence of the integral, including the positivity of the measure and the integrability of the jump sizes. The integral is split into negative and positive parts to handle potential discontinuities at zero. The key question is under what conditions the function I is well-defined, which Lévy addressed by providing sufficient conditions.Lévy processes are a class of continuous-time stochastic processes that combine Brownian motion and compound Poisson processes. Paul Lévy introduced the concept, leading to the Lévy-Khintchine formula, which describes a wide range of continuous and discontinuous stochastic processes. This chapter provides the fundamentals of Lévy processes essential for understanding Lévy option pricing models. Key references include Lévy (1954), Feller (1971), Bertoin (1996), and Sato (1999). Notations follow those in Chapter 16, which is recommended to be read before this chapter. The original construction of Lévy processes was heuristic, combining Brownian motion and a compensated compound Poisson process. The process $ X_t $ is defined as the sum of a Brownian motion $ X_t^{BM} $ and a compensated compound Poisson process $ X_t^{CP} $. The characteristic function of $ X_t $ is derived by combining the characteristic functions of the Brownian motion and the compound Poisson process, resulting in the de Finetti characteristic function. The exponent in the characteristic function is crucial. When combining multiple jump components, the Lévy measure $ \nu(dz) $ is introduced, which measures the intensity and distribution of jumps. The Lévy measure must satisfy certain conditions to ensure the integral exists. Lévy provided sufficient conditions for the existence of the integral, including the positivity of the measure and the integrability of the jump sizes. The integral is split into negative and positive parts to handle potential discontinuities at zero. The key question is under what conditions the function I is well-defined, which Lévy addressed by providing sufficient conditions.