In this article, simple tests of the random walk hypothesis are proposed that are robust against various kinds of conditional heteroskedasticity, non-stationarities, calendar effects and non-synchronous trading effects. In contrast, conventional tests are usually only robust against conditional heteroskedasticity. The robustness of the tests proposed in this paper is based on the fact that they examine the four popular summary measures (open, close, high, low) for each trading day separately. The results of a simulation study show that the tests are also quite robust against certain intraday anomalies like increased volatility at the beginning and at the end of the trading session. There is also evidence that the tests are robust against asymmetries in the returns.