Summary: We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the reference measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani [Probab. Theory Relat. Fields 137, No. 3–4, 541–593 (2007; Zbl 1113.60093)] with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a contracting Markov chain in 1-Wasserstein distance are also given. Throughout the text, several examples are worked out, including the cases of Gaussian measures on separable Banach spaces, and laws of diffusion processes.