The thesis has mainly two parts. The first part is the Econometric part, where we are trying to investigate the statistical properties of two methodologies that are used in the detection of Causality in Variance as well as in the Mean. Through several Monte Carlo simulations, we are trying to retrieve some conclusions concerning the empirical performance of these methodologies, using two Data Generating Processes, the GARCH and the FIGARCH model specifications. These methodologies are based on the estimation of the Cross Correlation Function for the squared standardised residuals. These residuals are obtained from the estimation of the univariate GARCH. The second part involves our empirical application. In this part we are trying to investigate the volatility as well as the return spillover among eighteen (18) countries of European Union (EU), based on the two methodologies, using both the GARCH and FIGARCH models.