Let $X_1, X_2, \ldots$ be iid random variables with common distribution $\gamma$, the standard Gaussian distribution on $\mathbb {R}$, and let $\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, be the empirical measures based on the sample $X_1, X_2, \ldots$. Is there anything known on the rate of convergence of $E (W_p (\mu_n, \gamma))$ as $n \to \infty$, where $W_p$ is the Wassertein distance with exponent $p \geq 1$? Same question if the $X_i$'s are standard Gaussian in $\mathbb{R}^k$.