This paper has two themes. First, we classify some effects which outliers in the data have on unit root inference. We show that, both in a classical and a Bayesian framework, the presence of additive outliers moves 'standard' inference towards stationarity. Second, we base inference on an independent Student-t instead of a Gaussian likelihood. This yields results that are less sensitive to the presence of outliers. Application to several time series with outliers reveals a negative correlation between the unit root and degrees of freedom parameter of the Student-t distribution. Therefore, imposing normality may incorrectly provide evidence against the unit root.