We survey recently introduced timelike curvature-dimension conditions for measured Lorentzian spaces which, using optimal transport means, are formulated by convexity properties of the Rényi entropy along chronological geodesics of probability measures. These constitute Lorentzian analogues of the (reduced) CD condition of Sturm and Bacher-Sturm for metric measure spaces, and complement the recent entropic approach by Cavalletti-Mondino, who use the Boltzmann entropy, after Erbar-Kuwada-Sturm. We discuss basic properties of (and relations between) these conditions, such as compatibility with the smooth case, stability, equivalences, sharp geometric inequalities, etc. Finally, we outline various possible directions of future research about spaces with synthetic timelike Ricci curvature bounds.