The most extensively studied form of credit risk is the default risk which is the risk that an obligor does not honour his payment obligations. Therefore, the main tool in the area of credit risk modelling is a sound specification of the random time of default. In this thesis, we investigate different ways to model the default time of a given reference entity. We do this by going beyond the existing standard reducedform models mostly characterized by their weakness to reproduce market credit spreads in terms of volatility featured and perfect calibration. We then address these problems as follows. First, we allow a two-sided jump to the default intensity process which gives an appealing solution but solves partially the calibration problem because it doesn’t fully preserve the range of the initial intensity process when forcing a perfect fit to market data. Secondly, we adopt a time change approach allowing for exact calibration to market survival probability curves without affecting the range of the original intensity process. Third, we consider a more general setup where we still preserve a reduced-form setting but under which the default time follows a structural approach with partial information flow. The last essay considers a direct modelling of the conditional survival probability associated to a default time. This approach is a case of “no-immersion” which is more convenient for application to credit derivatives which are characterized by strong wrong-way risk and high credit spreads.