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