  The metric $d(A,B)=\left[ \tr\, A+\tr\,
B-2\tr(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$ on the manifold of $n\times n$
positive definite matrices arises in various optimisation problems, in quantum
information and in the theory of optimal transport. It is also related to
Riemannian geometry. In the first part of this paper we study this metric from
the perspective of matrix analysis, simplifying and unifying various proofs.
Then we develop a theory of a mean of two, and a barycentre of several,
positive definite matrices with respect to this metric. We explain some recent
work on a fixed point iteration for computing this Wasserstein barycentre. Our
emphasis is on ideas natural to matrix analysis.
