Abstract: This study investigated whether the Nigerian stock market (from the period 1984 to 2006) follows a random walk. To carry out the investigation, the Generalized Autoregressive Conditional Hetrosecedasticity (GARCH) was employed. The result shows that the Nigerian stock market follows a random walk and is therefore weak form efficient. However, the years 1987, the period of financial deregulation, 1988 when some public companies were privatized, 1995 the period of internationalization of the Nigerian capital market and the years 2000-2006 recorded persistent volatility clustering suggesting weak form inefficiency in the market for these years. Nevertheless, the parameter estimates of their conditional mean equations (except in 1995) were insignificant. Besides these years, other years were conspicuously and significantly found to exhibit weak form efficiency. Thus, the Nigerian stock market is weak form efficient and as such no investor can usurp any privileged information to bit the market and make abnormal profit.