Stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets have attracted the attention of researchers since a long time. These are a set of properties, common across many instruments, markets and time periods that has been observed by independent studies. The objective of this research is to study stylized facts of financial time series by using data from the Indian financial market. BSE SENSEX is used as a proxy for the Indian market. Particular stress is given to the study of volatility using different models from the GARCH School including GARCH, EGARCH, TARCH, Asymmetric Component GARCH etc. The raw SENSEX daily return series are found to be non-normal having fat tails, show significant amount of asymmetry and exhibit strong volatility persistence as well as volatility clustering. This study also examines the possibility of long-term dependence (long memory) in Absolute SENSEX daily return series. Rescaled range analysis, modified rescaled range analysis and GPH test are used for this purpose. The results indicate presence of long memory and also show the series to be fractionally integrated. The findings obtained are in broad agreement with the stylized facts observed in financial time series.