The two main approaches to modeling defaults, structural and intensity based, will be reviewed. We show that perturbation methods are useful in approximating default probabilities in the context of stochastic volatility models. We then consider the case of many names and we discuss various ways of creating correlation of defaults. In highly-dimensional models, Monte Carlo simulations remain a powerful tool for computing prices of credit derivatives such as CDO's tranches and associated greeks. We propose an interactive particle system approach for computing the small probabilities involved in these financial instruments.