  Curvature is a key notion in General Relativity, characterizing the local
physical properties of spacetime. By contrast, the concept of curvature has
received scant attention in nonperturbative quantum gravity. One may even
wonder whether in a Planckian regime meaningful notions of (quantum) curvature
exist at all. Remarkably, recent work in quantum gravity using Causal Dynamical
Triangulations (CDT) has demonstrated both the existence and usefulness of a
new notion of quantum Ricci curvature (QRC), which relies neither on smooth
structures nor on tensor calculus. This overview article recalls some classical
notions related to curvature and parallel transport, as well as previous
unsuccessful attempts to construct quantum curvature observables based on
deficit angles and Wilson loops. It introduces the quasi-local QRC on piecewise
flat triangulations, and describes its behaviour in a purely classical setting,
its use in quantum observables, and currently known results in (C)DT quantum
gravity in two and four dimensions. The QRC opens the door to a range of
interesting physical observables that were previously out of reach, and will
help to bridge the gap between the nonperturbative quantum theory and
gravitational phenomena at lower energies.
