Summary In the study of asset returns, the preponderance of empirical evidence finds that return distributions are not normally distributed. Despite this evidence, non-normal multivariate modelling of asset returns does not appear to play an important role in asset management or risk management because of the complexity of estimating multivariate non-normal distributions from market return data. In this paper, we present a new subclass of generalized elliptical distributions for asset returns that is sufficiently user friendly, so that it can be utilized by asset managers and risk managers for modelling multivariate non-normal distributions of asset returns. For the distribution we present, which we call the multi-tail generalized elliptical distribution, we (1) derive the densities using results of the theory of generalized elliptical distributions and (2) introduce a function, which we label the tail function, to describe their tail behaviour. We test the model on German stock returns and find that (1) the multi-tail model introduced in the paper significantly outperforms the classical elliptical model and (2) the hypothesis of homogeneous tail behaviour can be rejected.