  Metric measure spaces with synthetic Ricci bounds have attracted great
interest in recent years, accompanied by spectacular breakthroughs and deep new
insights. In this survey, I will provide a brief introduction to the concept of
lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate
some of its geometric, analytic and probabilistic consequences, among them
Li-Yau estimates, coupling properties for Brownian motions, sharp functional
and isoperimetric inequalities, rigidity results, and structural properties
like rectifiability and rectifiability of the boundary. In particular, I will
explain its crucial interplay with the heat flow and its link to the
curvaturedimension condition formulated in functional-analytic terms by
Bakry-\`Emery. This equivalence between the Lagrangian and the Eulerian
approach then will be further explored in various recent research directions:
i) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for
singular spaces, ii) second order calculus, upper Ricci bounds, and
transformation formulas, iii) distribution-valued Ricci bounds which e.g. allow
singular effects of non-convex boundaries to be taken into account.
