This thesis focuses on examining potential impacts that structural breaks impose on volatility modelling via GARCH models. After incorporating structural breaks detected by the modified ICSS of Sanso et al. (2004) into conventional GARCH models, reduced volatility persistence is obtained for stock and foreign exchange returns in both China and the UK. A unidirectional volatility spillover is found going from stock to foreign exchange market for both countries, and ignoring structural breaks can lead to biased spillover patterns. These findings are well supported by comprehensive Monte Carlo simulations.