For X p-dimensional with expectation \(\mu\) and covariance matrix \(\Sigma\), \textit{J. B. S. Haldane}'s [Biometrika 35, 414-415 (1948; Zbl 0032.03601)] definition of a multivariate median as a minimizer \(\theta_ 0\) of \(E\| X-\theta \|\), where \(\| X-\theta\| = \{(X-\theta)'\Sigma^{-1}(X-\theta)\}^{1/2},\) is considered. It is generalized by studying the minimization of \(\Lambda(\theta)=E\Psi(X,\theta)\), where \(\Psi (X,\theta)=\Psi (\| X-\theta \|)\) and \(\psi\) is a real function on \([0,\infty)\) satisfying certain conditions. A (generalized) sample median is a minimizer \(\theta_ n\) of \({\bar \Psi}_ n(\theta)=n^{-1}\sum^{n}_{i=1}\Psi_ n(X_ i,\theta),\) where \(\Psi_ n\) is defined by replacing \(\Psi\) in \(\Sigma\) by a consistent estimator \(\Sigma_ n.\) Assuming that the domain of \(\theta\) is a compact subset of \({\mathbb{R}}^ p\) and that \(\bigwedge (\theta)\) has a unique minimizer \(\theta_ 0\), convergence in probability of \(\theta_ n\) to \(\theta_ 0\) is proved. Under further assumptions asymptotic normality of \((\theta'_ n,\bar X'_ n, \text{vec}(\Sigma_ n)')'\) is proved. Making use of the median, three types of measures of multivariate skewness are introduced and their asymptotic null distributions are obtained. The measures are generalizations of proposals by Pearson, Geary and Rayleigh. A numerical illustration is provided.