A problem with the classical firm value model of Merton (1974) arises from modeling the firm value in terms of a diffusion. The resulting term structure of the credit spreads slopes upwards from zero, even for financially stable firms, implying that their default risks are increasing with time. In reality credit spread curves can also slope downwards or be flat. Another issue is the expectancy of a default: With diffusion models, one has an increasing sequence of stopping times converging towards the default time. A firm can therefore never default unexpectedly with this approach. It is not possible for neither structural nor intensity based models based on diffusions to model both expected and unexpected defaults. The incorporation of jump-diffusions has been shown to generate the correct shapes of the yield spread curves and match the sizes of the credit spreads of corporate bonds. Furthermore, the possibility of an unexpected default of the firm is also taken care of by the jumps in the credit risk. This thesis will be organized as follows: First, an introduction to the most basic concepts in stochastic analysis is given. The results are then utilized in the following chapters about modeling credit risk, where the theory of pricing and hedging of certain credit derivatives is presented. The need of including Levy processes will become evident, and an introduction is given. The Vasicek intensity model (for both diffusions and jump processes) is calibrated to market data in order to price both default-free and defaultable bonds. Finally, an extension of the Vasicek model to a regime-switched version is discussed (more specifically in the setting of bond pricing) and calibrated to market data.