  Let $\pi\in \Pi(\mu,\nu)$ be a coupling between two probability measures
$\mu$ and $\nu$ on a Polish space. In this article we propose and study a class
of nonparametric measures of association between $\mu$ and $\nu$, which we call
Wasserstein correlation coefficients. These coefficients are based on the
Wasserstein distance between $\nu$ and the disintegration $\pi_{x_1}$ of $\pi$
with respect to the first coordinate. We also establish basic statistical
properties of this new class of measures: we develop a statistical theory for
strongly consistent estimators and determine their convergence rate in the case
of compactly supported measures $\mu$ and $\nu$. Throughout our analysis we
make use of the so-called adapted/bicausal Wasserstein distance, in particular
we rely on results established in [Backhoff, Bartl, Beiglb\"ock, Wiesel.
Estimating processes in adapted Wasserstein distance. 2020]. Our approach
applies to probability laws on general Polish spaces.
