This paper studies the rate of convergence of the empirical measure $\mu_N$ to a target distribution $\mu$ in the Wasserstein distance of order $p > 0$. The authors provide non-asymptotic $L^p$-bounds and concentration inequalities for any $p > 0$ and $d \geq 1$. They extend these results to stationary $\rho$-mixing sequences, Markov chains, and interacting particle systems. The paper introduces the Wasserstein distance $W_p$ and analyzes the convergence of $\mu_N$ to $\mu$ in this metric. The main results include moment estimates and concentration inequalities for the Wasserstein distance between the empirical measure and the target distribution. The authors derive bounds on the expected value of the Wasserstein distance and show that it concentrates around its mean with high probability. The paper also discusses the relationship between the dimension $d$, the cost parameter $p$, and the moment conditions on the target distribution $\mu$. It provides examples and lower bounds to illustrate the sharpness of the results. The authors use a coupling approach to derive the bounds and show that the Wasserstein distance can be controlled by a related distance $\mathcal{D}_p$. The paper concludes with a plan of the paper, outlining the proofs of the main results and the techniques used to derive the bounds. The authors also discuss the implications of their results for various applications, including quantization, optimal matching, density estimation, and MCMC methods.This paper studies the rate of convergence of the empirical measure $\mu_N$ to a target distribution $\mu$ in the Wasserstein distance of order $p > 0$. The authors provide non-asymptotic $L^p$-bounds and concentration inequalities for any $p > 0$ and $d \geq 1$. They extend these results to stationary $\rho$-mixing sequences, Markov chains, and interacting particle systems. The paper introduces the Wasserstein distance $W_p$ and analyzes the convergence of $\mu_N$ to $\mu$ in this metric. The main results include moment estimates and concentration inequalities for the Wasserstein distance between the empirical measure and the target distribution. The authors derive bounds on the expected value of the Wasserstein distance and show that it concentrates around its mean with high probability. The paper also discusses the relationship between the dimension $d$, the cost parameter $p$, and the moment conditions on the target distribution $\mu$. It provides examples and lower bounds to illustrate the sharpness of the results. The authors use a coupling approach to derive the bounds and show that the Wasserstein distance can be controlled by a related distance $\mathcal{D}_p$. The paper concludes with a plan of the paper, outlining the proofs of the main results and the techniques used to derive the bounds. The authors also discuss the implications of their results for various applications, including quantization, optimal matching, density estimation, and MCMC methods.