  This is a review of the ideas behind the Fisher--Rao metric on classical
probability distributions, and how they generalize to metrics on density
matrices. As is well known, the unique Fisher--Rao metric then becomes a large
family of monotone metrics. Finally I focus on the Bures--Uhlmann metric, and
discuss a recent result that connects the geometric operator mean to a geodesic
billiard on the set of density matrices.
