We propose a posterior odds analysis of the hypothesis of a unit root in real exchange rates. From a Bayesian viewpoint the random walk hypothesis for real exchange rates is a posteriori as probable as a stationary AR(1) process for four out of eight time series investigated. The French franc / German Dmark is clearly stationary, while the Japanese yen / US dollar is most likely a random walk.' In contrast, classical tests are unable to reject the unit root for any of these series.