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Let $\mu_N$ be the empirical measure associated to a $N$-sample of a givenprobability distribution $\mu$ on $\mathbb{R}^d$. We are interested in the rateof convergence of $\mu_N$ to $\mu$, when measured in the Wasserstein distanceof order $p>0$. We provide some satisfying non-asymptotic $L^p$-bounds andconcentration inequalities, for any values of $p>0$ and $d\geq 1$. We extendalso the non asymptotic $L^p$-bounds to stationary $\rho$-mixing sequences,Markov chains, and to some interacting particle systems.

On the rate of convergence in Wasserstein distance of the empirical measure