We introduce the directionally dispersed class of multivariate distributions, a generalization of the elliptical class. By allowing dispersion of multivariate random variables to vary with direction it is possible to generate a very wide and flexible class of distributions. Directionally dispersed distributions have a simple form for their density, which extends a spherically symmetric density function by including a function D modelling directional dispersion. Under a mild condition, the class of distributions is shown to preserve both unimodality and moment existence. By adequately defining D, it is possible to generate skewed distributions. Using spline models on hyperspheres, we suggest a very flexible, yet practical, implementation for modelling directional dispersion in any dimension. Finally, we use the new class of distributions in a Bayesian regression set-up and analyse the distributions of a set of biomedical measurements and a sample of US manufacturing firms.