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This paper examines the time series behavior of monthly bilateral real exchange rates (RER) on a comprehensive sample of 78 industrialized and developing countries, using the US Dollar, the UK Pound and the German Deutsche Mark as numeraires. We suggest a three step testing procedure based on recently introduced econometric techniques, in order to assess the mean-reverting properties of the RER and to address the question of whether real exchange rates follow a non linear process or a long memory process. The main results are as follows. Firstly, most of the bilateral real exchange rates under study are not mean-reverting. Secondly, the nonlinear ESTAR type adjustment is far from being prominent. Finally, only few bilateral RER exhibit true long memory mean-reverting properties.