Multinomial choice models are used for the analysis of unordered, mutually exclusive choice alternatives. Conventionally used multinomial choice models are the multinomial logit, nested logit, multinomial probit and random parameters logit models. This paper develops multinomial choice models based on Archimedean copulas. In contrast to the multinomial logit and nested logit models, no independence of irrelevant alternatives property is implied. Moreover, copula-based multinomial choice models are more parsimonious than the multinomial probit and random parameters logit models, which makes them attractive from a computational point of view and makes them particularly suitable for prediction purposes. When the number of alternatives becomes large, additional complexity can be introduced using nested Archimedean copulas. Nested structures can often be motivated from individual behavior. The paper also considers an empirical application to travel mode choice. It is found that copula-based multinomial choice models provide a good compromise between fit and parsimony.