The Credit Default Swap (CDS) is a bilateral contract, that refers to the credit-worthiness of a ?reference entity?. A reference entity or ?reference obligor? typically is a corporation, a financial institution or a government. The CDS is a credit derivative, which makes trading with default risk of loans and bonds possible. The risk is transferred from the CDS-buyer to the CDS-seller. Credit and credit risk will be defined in chapter two. In return for risk transfer, the seller receives quarterly payments from the protection buyer. In Exchange, the buyer receives a payoff if the bond defaults. A default is often referred to as a ?credit event?. The events triggering a credit derivative are defined by the International Swaps and Derivatives Association (ISDA). The probability of default is accordingly the probability of occurrence of a credit event. This probability has to be quantified by credit risk models. This thesis focuses, besides the analysis of the CDS-market, on the mathematical derivation of these models. You have to distinguish between structural models and reduced-form models. Altogether, three representatives will be specified, the Merton-model, the Jarrow-Turnbull-model and the Jarrow-Lando-Turnbull-model. They build the basic models to many extensions and will be implemented as well as judged in the sixth chapter. For the market analysis, the author used data, which is provided by the ISDA, the Bank for International Settlements (BIS) and the Depository Trust and Clearing Corporation (DTCC).