This talk will be a review of the theory of weighted Lorentz-Finsler manifolds. A Lorentz-Finsler manifold is a generalization of a Lorentzian manifold in the same way that a Finsler manifold generalizes a Riemannian manifold. One can further equip a Lorentz-Finsler manifold with a time orientation as well as a weight, then we have a weighted Finsler spacetime. In this general framework, we can successfully develop the theory of Ricci curvature (singularity theorems, various comparison theorems, etc.). This is joint work (partly in progress) with Mathias Braun (Toronto), Yufeng Lu (Hong Kong), Ettore Minguzzi (Firenze).