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The unit root hypothesis is examined allowing a possible one-time change in the level or in the slope of the trend function. When fluctuations are stationary around a breaking trend function, standard tests cannot reject the unit root, even asymptotically. Consistent tests are derived and applied to the Nelson-Plosser data set (allowing a change in level for the 1929 crash) and to the postwar quarterly real GNP series (allowing a change in slope after 1973). The unit root hypothesis is rejected at a high confidence level for most series. Fluctuations are stationary. The only persistent "shocks" are the 1929 crash and the 1973 oil price shock. Copyright 1989 by The Econometric Society.