Barycentre is the geometric mean of a distribution on a metric space; it is a point that minimizes its average distance squared to the given distribution. Such a point is highly non-unique in general, though we have uniqueness when the underlying space is the Euclidean space or more generally a space of nonpositive curvature with trivial topology. We consider such a notion from the viewpoint of optimal transport which gives a natural distance structure between probability measures. 
This allows us to uniquely interpolate many probability distributions, called the Wasserstein barycentre, as initiated by Agueh and Carlier. It also leads to a uniquely defined canonical barycentre, which is obtained by relaxing the notion of barycentre point to a barycentre measure. We will explain these developments, based on join work with Brendan Pass.