Summary: A class of nonlinear processes which have a root that is not constant, but is stochastic, and varying around unity is introduced. The process can be stationary for some periods, and mildly explosive for others. Stochastic unit roots are see to arise naturally in economic theory, as well as in everyday macroeconomic applications. It is shown that standard tests, such as the augmented Dickey-Fuller test, cannot easily distinguish between exact unit roots and stochastic unit roots. An alternative test which has difference stationarity as the null suggests that exact unit-root models are often rejected in favor of more general nonlinear stochastic unit-root (STUR) models. Estimation is discussed, and, a forecast comparison of linear random walk and \(\text{AR} (p)\) models, time-varying parameter models, and STUR models suggests that this new class of processes is potentially useful, particularly when the objective is multi-step ahead forecasting.