Researchers in quantum gravity look for a theoretical and quantitative understanding of the dynamics of spacetime near the ultrashort Planck scale, which is governed by large quantum fluctuations and cannot be described by perturbation theory around solutions to the classical Einstein equations. A powerful nonperturbative methodology is that of "random geometry", the study of continuum limits of statistical ensembles of piecewise flat simplicial manifolds consisting of equilateral triangular building blocks. While systems of two-dimensional random geometry in Riemannian signature provide analytically soluble toy models and illustrate the viability of the method, physical quantum gravity requires an implementation in four dimensions and Lorentzian signature.

This has been achieved in the quantum gravity theory based on Causal Dynamical Triangulations (CDT), which has produced highly nontrivial evidence that its random-geometric ensemble leads to a well-defined quantum geometry, whose large-scale properties are compatible with a classical, four-dimensional de Sitter universe. Its physical properties are explored "experimentally" by measuring suitable quantum observables with the help of computer simulations. An exciting recent development is the introduction of a notion of (quantum) Ricci curvature -- inspired by the Ollivier-Ricci curvature -- which is applicable in the nonperturbative quantum regime and yields finite results. I will describe its geometric construction, its implementation on a variety of classical and quantum metric spaces, and the new window it has opened on the physics on quantum gravity.