This paper tests the random walk hypothesis for the log-differenced monthly US real exchange rates versus some major currencies. The tests we use are variance ratio test, Durlauf's (1991) spectral domain tests and Andrews and Ploberger's ( 1996) optimal tests. The variance ratio test is calculated by using Andrews' (1991) optimal data-dependent methods. Finite sample properties of these tests are also reported. Because the results of applying these tests to the real exchange rates are occasionally inconsistent, tests to synthesize these test results are proposed and applied to the real exchange rates. These tests have often been used in meta-analysis, but have not previously been used to synthesize different test results. Simulation results for these tests are also reported. For the real exchange rate data from the post-Bretton Woods period, these tests reject the null only for the Swiss franc. But when longer-horizon data are used, there is more evidence of serial correlations in the log-differenced real exchange rates.