Recent research has found that trend-break unit root tests derived from univariate linear models do not support the hypothesis of long-run purchasing power parity (PPP) for US dollar real exchange rates. In this paper univariate smooth transition models are utilized to develop unit root tests that allow under the alternative hypothesis for stationarity around a gradually changing deterministic trend function. These tests reveal statistically significant evidence against the null hypothesis of a unit root for the real exchange rates of a number of countries against the US dollar. However, restrictions consistent with long-run PPP are rejected for some of the countries for which a rejection of the unit root hypothesis is obtained.