AbstractMeasures of centrality that generalize the univariate median are studied and applied to multivariate and directional distributions. A standard example is developed for general multivariate settings, and the uniqueness of the median proved for distributions satisfying certain regularity conditions. In the presence of weaker regularity, this median is shown to be of codimension 2. Conditions are also provided for these measures of centrality to be equivariant under transformations on the sample space. The equivariance of the usual univariate median under monotone transformations is seen as a special case.