This paper explores a general procedure to perturb a multivariate density that satisfies a weak form of multivariate symmetry, generating a set of non-symmetric densities. The approach is broad enough to encompass various recent proposals related to the skew normal distribution. The paper focuses on skew elliptical densities and establishes connections with existing work. It specializes to a form of multivariate skew t density, examining likelihood inference and providing numerical examples. The skew t distribution is defined as a transformation of a skew normal variate and a chi-square variate, leading to a density that is mathematically manageable and flexible in skewness and kurtosis. The paper discusses the properties of this distribution, including its moments and linear and quadratic forms, and provides algorithms for computing its distribution function.This paper explores a general procedure to perturb a multivariate density that satisfies a weak form of multivariate symmetry, generating a set of non-symmetric densities. The approach is broad enough to encompass various recent proposals related to the skew normal distribution. The paper focuses on skew elliptical densities and establishes connections with existing work. It specializes to a form of multivariate skew t density, examining likelihood inference and providing numerical examples. The skew t distribution is defined as a transformation of a skew normal variate and a chi-square variate, leading to a density that is mathematically manageable and flexible in skewness and kurtosis. The paper discusses the properties of this distribution, including its moments and linear and quadratic forms, and provides algorithms for computing its distribution function.