  Borel probability measures living on metric spaces are fundamental
mathematical objects. There are several meaningful distance functions that make
the collection of the probability measures living on a certain space a metric
space. We are interested in the description of the structure of the isometries
of such metric spaces. We overview some of the recent results of the topic and
we also provide some new ones concerning the Wasserstein distance. More
specifically, we consider the space of all Borel probability measures on the
unit sphere of a Euclidean space endowed with the Wasserstein metric $W_p$ for
arbitrary $p \geq 1,$ and we show that the action of a Wasserstein isometry on
the set of the Dirac measures is induced by an isometry of the underlying unit
sphere.
