In the framework of risk assessment, computer codes are increasingly used to understand, model and predict physical phenomena. As these codes can be very time-consuming to run, which severely limit the number of possible simulations, a widely accepted approach consists in approximating the CPU-time expensive computer model by a so-called "surrogate model". In this context, the Gaussian Process regression (also called kriging) is one of the most popular technique. It offers the advantage of providing a predictive distribution for all new evaluation points. An uncertainty associated with any quantity of interest (e.g., a probability of failure in reliability studies) to be estimated can thus be deduced and adaptive strategies for choosing new points to run with respect to this quantity can be developed. This paper focuses on the estimation of the Gaussian process covariance parameters by reviewing recent works on the analysis of the advantages and disadvantages of usual estimation methods, the most relevant validation criteria (for detecting poor estimation) and recent robust and corrective methods.