Investors are constantly asking whether beating the market on a consistent basis is possible. There is probably no definitive answer to the question of how to make a guaranteed profit (or return) because index prices can fluctuate at any time. The aim of most investors, therefore, is to predict the stock market return and the volatility, (a measure of investment nsk) and this requires an understanding of stock market behaviour. In this research, diierent techniques, both previously existing and newly developed here (and associated specifically with the discrete wavelet transform (DWT)), are applied to study the behav~our of global stock market indices We consider type of memory, mterrelationships between stock markets, market reaction to crashes and events, and the best indicators of market types (short-term, long-term or mixed). The unifylng aim is to provide a baseline set of characteristic features which typify behaviors of given market type Principal remarks include the fact that the DWT, alone or with other methods, can succeed in providing an in-depth view of these data, in particular when confronted with non-stationary, non-normal and noisy characteristics. The approach provides an important method for the aualysis and interpretation of financial market time series. Our principal findings on volatility measures, moreover, show strong evidence of long-term memory effects, which are not evident in the returns themselves. Emerging and Mature markets are found to deal differently with crashes and events with the latter taking a shorter time to recover from crises on average, compared to the former. Furthermore, we conclude that this binary classification is too simple and stock markets can now be demonstrated to fall into more than two groups, with the designation L'emerging" ("developing") and "mature" ("developed") proving imprecise. Additionally, and in the context of the global market, from Chapter 5, we note that international co-movements and volatility (or nsk) have increased markedly since the middle of the 20th century and that cloclnuzse transmtssion between global stock markets is observed, i.e from Asaa to h o p e to Amerzca back to Asia). The combination of ~nternadl ependencies and external influences provide the impacts for stock market volatility. The ultimate goal, of course, would be to anticipate these Impacts to be able to make the rlght investment decision.