In common with most forms of designed experiment, crossover trials can be affected by missing data. Attempts to devise designs that can mitigate the possible effects of missing data, such as loss of efficiency, or even inestimability of certain contrasts, have been proposed. However, a potentially serious effect of missing data that has not been addressed in designs hitherto is that the treatment effects may be biassed because of the nature of the missingness process. We investigate this problem in two-treatment, two-period crossover designs. In particular, we consider the robustness of the analysis under a missing at random assumption when, in fact, the data are non-ignorably missing. We show that the conventional AB/BA design still has good properties, although the design with sequences AB, BA, AA, and BB may be preferred if the chance of dropout depends primarily on the difference between the responses in the two periods.