In this thesis we survey GARCH modelling with special focus on the fitting of GARCH models to financial return series. The robustness of the estimation of the parameters in the model is examined with three different distributional assumptions for the innovations; Gaussian distribution, Student-t distribution and GED (Generalised Error Distribution). Both the Student-t distribution and the GED have fat tails. The maximum-likelihood approach is used for the parameter estimation. Using backtesting, the related residuals under the three different distributional assumptions are examined.