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Outliers in discrete choice response data may result from misclassification and misreporting of the response variable and from choice behaviour that is inconsistent with modelling assumptions (e.g. random utility maximisation). In the presence of outliers, standard discrete choice models produce biased estimates and suffer from compromised predictive accuracy. Robust statistical models are less sensitive to outliers than standard non-robust models. This paper analyses two robust alternatives to the multinomial probit (MNP) model. The two models are robit models whose kernel error distributions are heavy-tailed t-distributions to moderate the influence of outliers. The first model is the multinomial robit (MNR) model, in which a generic degrees of freedom parameter controls the heavy-tailedness of the kernel error distribution. The second model, the generalised multinomial robit (Gen-MNR) model, is more flexible than MNR, as it allows for distinct heavy-tailedness in each dimension of the kernel error distribution. For both models, we derive Gibbs samplers for posterior inference. In a simulation study, we illustrate the excellent finite sample properties of the proposed Bayes estimators and show that MNR and Gen-MNR produce more accurate estimates if the choice data contain outliers through the lens of the non-robust MNP model. In a case study on transport mode choice behaviour, MNR and Gen-MNR outperform MNP by substantial margins in terms of in-sample fit and out-of-sample predictive accuracy. The case study also highlights differences in elasticity estimates across models.