With the increased interest in the African economy, it is vital that we measure the performance of our capital markets to know where they stand. The Efficient Market Hypothesis (EMH) seeks to test whether a stock market is efficient in either the weak, semi-strong or strong form. With Kenya being an emerging market, the weak form efficient market hypothesis was put to test by the researcher, by determining whether successive daily stock market returns on the Nairobi Securities Exchange follow a random Walk or otherwise. The EMH briefly argues that for an efficient market, future share prices and returns should be random and unpredictable, such that any information regarding a stock is quickly assimilated into the market to reflect on the new share price. Data in the form of historical daily closing NSE 20-share Index from 1st January 2008 to 31st December 2012 was obtained from the Nairobi Securities Exchange. The use of a longer time period was to eliminate the thin trading bias that is characteristic of emerging stock markets, while the use of indices is to maintain consistency of data used in the research. Both parametric and non-parametric tests were used, to confirm results obtained in either of the tests. The data was analysed using STATA statistical package to test for stationarity of the model, normal distribution of stock prices, randomness of successive price changes and independence of stock price changes. Unit root test, runs test and Autocorrelation tests were carried out to test for the afore mentioned characteristics of the stock price and returns. Mixed results were obtained from the research, with the runs test concluding that the NSE daily market return series was random and therefore the NSE followed the random walk model. The autocorrelation tests and unit root tests, however, concluded the NSE was not weak form efficient. The autocorrelation tests detected serial correlation in the successive daily market returns and there was absence of a unit root in the time NSE time series. The research concluded that the NSE was not weak form efficient, since all the tests conducted did not conform to the characteristics of weak form efficient market hypothesis. Information flow from the listed companies to the public is not efficient, giving some investors an advantage over others. It was recommended after the study that the NSE should put policies in place to ensure informational efficiency and also educate the public on the advantages of investing in the stock market to improve trading on the bourse.