In this chapter, diffusion and jump-diffusion processesJump-diffusion process are discussed. Diffusion processes are continuous parameter continuous state processes with sample paths that are continuous everywhere but differentiable nowhere. The most common diffusion processDiffusion process is Brownian motionBrownian motion (aka Wiener process). Brownian motion is a continuous parameter continuous state stochastic process having stationary and independent increments. Diffusion processes can be a function of Brownian motion or integral with respect to Brownian motion. In this chapter, the construction of the Brownian motion and its various properties are presented. The main discussion will be about some of the important processes derived from Brownian motion and their properties. Further, this chapter introduces jump diffusion processes. Jump diffusion processes are the stochastic processes that involve jumps and diffusion components. The fundamental pure jump process is the Poisson processPoisson process which serves as starting point for jump processes. The jumps of Poisson process are of fixed size one. Another important jump process which is widely used is compound Poisson processPoisson processcompound. Compound Poisson process can be seen as a generalized version of Poisson process with jump size as value of a RV. It is similar to Poisson process, but its jumps are of random size. This chapter also introduces Lévy processes. Lévy processes are processes with independent and stationary increments. Brownian motion, Poisson processes and compound Poisson process are examples of Lévy processes and they also form building blocks for Lévy processes. The general properties of Lévy processes, Lévy-Khintchin formula and Lévy Itô decomposition will also be discussed. Lévy-Khintchin formula helps to study distributional properties of Lévy processes and Lévy Itô decomposition supports in visualization of sample paths of Lévy processes. At the end, a special type of jump process is introduced named Hawkes jump processHawkes jump process. Hawkes process is a self exciting point process which takes into consideration the clustering of jumps phenomena. In this process when a jump occurs in the stochastic process, it increases the probability of the occurrence of future jumps, thus resulting in jump clustering. In Poisson jump processes, due to independent increments, this self exciting behaviour is not observed.