This paper proposes two new unit root tests that are appropriate in the presence of an unknown number of structural breaks in the level of the data. One is based on a single time series and the other is based on a panel of multiple series. For the estimation of the number of breaks and their locations, a simple procedure based on outlier detection is proposed. The limiting distributions of the tests are derived and evaluated in small samples using simulation experiments. The implementation of the tests is illustrated using as an example purchasing power parity.