In this talk I will take some care to introduce the general class of Levy processes as well as the most relevant parametric families. I will explain how these can be used for modelling purposes, directly or as driving processes for more general stochastic processes. As an application, I'll discuss stochastic volatility modelling and some questions arising when doing inference in the presence of jumps, based on joint work with Ole Barndorff-Nielsen and Neil Shephard.