A fairly general procedure is studied to perturbate a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is general enough to encompass a number of recent proposals in the literature, variously related to the skew normal distribution, The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew-t density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.