We study general risk models where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a Lévy process into the model. This seeks to account for the discrete nature of claims and asset prices. Over the years, we sudied several explicit examples of Lévy processes that can be used to drive a risk model. We have also studied important features of such processes and their relevance to risk modeling. Such processes are now known as Lévy Insurance Risk Models.