  Measures of central tendency such as mean and median are a primary way to
summarize a given collection of random objects. In the field of optimal
transport, the Wasserstein barycenter corresponds to Fr\'{e}chet or geometric
mean of a set of probability measures, which is defined as a minimizer of the
sum of squared distances to each element in a given set when the order is 2. We
present the Wasserstein median, an equivalent of Fr\'{e}chet median under the
2-Wasserstein metric, as a robust alternative to the Wasserstein barycenter. We
first establish existence and consistency of the Wasserstein median. We also
propose a generic algorithm that makes use of any established routine for the
Wasserstein barycenter in an iterative manner and prove its convergence. Our
proposal is validated with simulated and real data examples when the objects of
interest are univariate distributions, centered Gaussian distributions, and
discrete measures on regular grids.
