Let $μ_N$ be the empirical measure associated to a $N$-sample of a given probability distribution $μ$ on $\mathbb{R}^d$. We are interested in the rate of convergence of $μ_N$ to $μ$, 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 $ρ$-mixing sequences, Markov chains, and to some interacting particle systems.