This study presents various GARCH models for predicting movements (returns) and volatility patterns in major national stock market indices. These models depict that future returns in the national stock markets of Australia, Belgium, Canada, France, Italy and Switzerland are predictable from past realized returns; therefore, the stock indices for these markets violate the Martingale model. The stock market indices for Germany, Japan, the United Kingdom and the United States, however, behave as Martingale processes. Volatility patterns in these indices are predictable using past realized volatility measures.