  Wasserstein distances are metrics on probability distributions inspired by
the problem of optimal mass transportation. Roughly speaking, they measure the
minimal effort required to reconfigure the probability mass of one distribution
in order to recover the other distribution. They are ubiquitous in mathematics,
with a long history that has seen them catalyse core developments in analysis,
optimization, and probability. Beyond their intrinsic mathematical richness,
they possess attractive features that make them a versatile tool for the
statistician: they can be used to derive weak convergence and convergence of
moments, and can be easily bounded; they are well-adapted to quantify a natural
notion of perturbation of a probability distribution; and they seamlessly
incorporate the geometry of the domain of the distributions in question, thus
being useful for contrasting complex objects. Consequently, they frequently
appear in the development of statistical theory and inferential methodology,
and have recently become an object of inference in themselves. In this review,
we provide a snapshot of the main concepts involved in Wasserstein distances
and optimal transportation, and a succinct overview of some of their many
statistical aspects.
