  The goal of the present work is three-fold.
  The first goal is to set foundational results on optimal transport in
Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of
optimal couplings and Kantorovich duality (several results are new even for
smooth Lorentzian manifolds).
  The second one is to give a synthetic notion of ``timelike Ricci curvature
bounded below and dimension bounded above'' for a measured Lorentzian
pre-length space using optimal transport. The key idea being to analyse
convexity properties of Entropy functionals along future directed timelike
geodesics of probability measures. This notion is proved to be stable under a
suitable weak convergence of measured Lorentzian pre-length spaces, giving a
glimpse on the strength of the approach we propose.
  The third goal is to draw applications, most notably extending volume
comparisons and Hawking singularity Theorem (in sharp form) to the synthetic
setting.
  The framework of Lorentzian pre-length spaces includes as remarkable classes
of examples: space-times endowed with a causally plain (or, more strongly,
locally Lipschitz) continuous Lorentzian metric, closed cone structures, some
approaches to quantum gravity.
