  We investigate the notion of Wasserstein median as an alternative to the
Wasserstein barycenter, which has become popular but may be sensitive to
outliers. In terms of robustness to corrupted data, we indeed show that
Wasserstein medians have a breakdown point of approximately $\frac{1}{2}$. We
give explicit constructions of Wasserstein medians in dimension one which
enable us to obtain $L^p$ estimates (which do not hold in higher dimensions).
We also address dual and multimarginal reformulations. In convex subsets of
$\mathbb{R}^d$, we connect Wasserstein medians to a minimal (multi) flow
problem \`a la Beckmann and a system of PDEs of Monge-Kantorovich-type, for
which we propose a $p$-Laplacian approximation. Our analysis eventually leads
to a new numerical method to compute Wasserstein medians, which is based on a
Douglas-Rachford scheme applied to the minimal flow formulation of the problem.
