The concept of transportation distance between attractors in dynamical systems allows one to express how closely the long-term behaviour of two given systems resemble each other. This is a particular example of a Wasserstein distance between probability measures. Wasserstein distances are more robust than other distances and have interesting theoretical features, but, on the other hand, they are more difficult to compute, so that one usually has to resort to various approximations. In the paper under review, the authors show how these distances can be analyzed statistically to provide useful qualitative and quantitative information about the global structure of dynamical systems. In particular, it is shown that they can be used to classify and discriminate time series, to detect and quantify synchronization phenomena between different dynamical systems and characterize and track bifurcations when the parameters of the system change. The basic properties of the Wasserstein distances are illustrated by the well-known Hénon map. The methodology developed here is subsequently applied to real problems such as a data set of tidal breathing records to discriminate between patients suffering from asthma and those suffering from chronic obstructive pulmonary disease, as well as time series arising from magneto-encephalografic recordings to detect and quantify functional connectivities.