What should a meaningful notion of curvature satisfy, in the absence of a smooth structure? There are many generalizations of various curvatures to non-smooth metric spaces (e.g. Ollivier's Ricci curvature). Suppose I have a metric space $(X,d)$ and I want to define a notion of curvature that resembles some classical curvature. What are some properties that this curvature could have in order to qualify as a generalization? For instance, one answer could be that a curvature on a metric measure space $(X,d,\mu)$ should admit an analog of the Levy-Gromov isoperimetric inequality, i.e. positive curvature leads to a control on the isoperimetry of balls in $(X,d,\mu)$.
 A: A curvature bound for a non-smooth metric space is often known as a "synthetic curvature bound" or a "coarse curvature." Here's one possible definition for the concept.

Definition: A condition $Q_{\kappa}$ is a synthetic lower bound for a curvature tensor $R$ if the following two conditions hold:
(1) On a smooth manifold $M$ where $R$ is defined,
$$ R \geq \kappa \Longleftrightarrow Q_{\kappa} $$
(2) The condition $Q_{\kappa}$ is well-defined for spaces with low regularity (for which the tensor $R$ may not be well-defined).

It's worth noting that there are many ways to define synthetic curvature bounds, and it's even possible to find formulations which are inequivalent (although you do want them to agree for Riemannian manifolds). As such, the right way to do it will depend on the context, so you generally want to have some application in mind. Let me describe a few examples to give some intuition.
Synthetic sectional curvature
One good way to find synthetic curvature bounds is to look for results in Riemannian (or pseudo-Riemannian, Kahler, etc.) geometry that hold if and only if the curvature satisfies some bound. Then, one can simply define a metric (or length) space to satisfy $Q_{\kappa}$ if it satisfies the conclusion of the theorem. For instance, if one uses this idea with the Topogonov triangle comparison theorem, one arrives at the definition of an Alexandrov space. Analogously, one can use this idea to define upper bounds for the curvature (which yields the idea of a $CAT(\kappa)$ space.) Both of these give synthetic notions of sectional curvature bounds.
Synthetic Ricci curvature
It's a bit harder to define synthetic Ricci curvature bounds, since the usual definition involves either computing the trace or average value of sectional curvatures, both of which are problematic for non-smooth or infinite dimensional spaces. However, if one considers metric spaces with an induced measure, i.e., metric-measure spaces, one can define synthetic Ricci curvature in terms of the displacement interpolation on the space. Ollivier's approach to synthetic Ricci curvature is along these lines, but let me mention another approach by Lott and Villani.
For this, one considers two distributions of gas and studies the flow between them which minimizes some cost function which is induced by a Lagrangian. In a space with negative Ricci curvature, the particles will be squeezed together as the travel from the initial to final distribution, and the reverse will occur on a space of positive Ricci curvature (as shown below). To make this idea precise, one studies the convexity of the entropy along displacement interpolation, and this leads to the notion of $N$-Ricci curvature bounds.

