This study extends evidence on the efficiency of stock markets in developing countries using data from the Nairobi Stock Exchange (NSE). Previous evidence from studies on stock markets in developing countries, and NSE in particular, is inconclusive. In many cases, the findings have not supported the random walk hypothesis and are therefore not consistent with efficiency in the weak-form. The key question investigated is whether successive share price returns on the Nairobi Stock Exchange are independent random variables so that price returns cannot be predicted from historical price returns. This study uses the traditional random walk methodology of serial correlation and runs tests as applied by Fama (1965), Cooper (1982), and Taylor (1986) rather than the newer methodologies of variance ratios [Lo and MacKinlay (1988)] and of regression [Jegadeesh (1990)]. These techniques are used for reasons of triangulation in research and for their intuitive appeal. They remain appropriate tools for testing the weak-form EMH despite challenge from newer methodologies. In their use, nevertheless, the study recognises and deals with two largely ignored issues in their application to EMH tests in emerging markets: the quality and quantity of data, and the depth of analysis of the market microstructure. The quality and quantity of data are improved through the creation of a computer database. The study then analyses all three price series on the exchange: The Bid, Ask and Transaction prices. The findings suggests that with proper control over the quality of the data and the use of a larger number of data observations, the random walk model can be a good description of successive price returns in an emerging stock market. This has been shown to hold irrespective of whether bid, ask, or transaction returns are used. This is contrary to most of the earlier evidence that the random walk model does not apply in such markets. The results obtained are therefore consistent with the weak-form of the EMH.