Using a notion of Ricci curvature for Markov chains on metric spaces, we give in this talk non-asymptotic variance bounds (resp. concentration inequalities) for the empirical mean of Markov chains. The main ingredient in the proofs relies on a tensorization procedure of the variance (resp. the Laplace transform) and allows us to consider, by a limiting argument, either continuous-time Markov chains with unbounded generator or diffusion processes.