Motivated by geometric (and also physical) problems, Misha Gromov advocated
the loosening of the relation between a metric and a measure that is
familiar in Riemannian geometry where the metric is obtained by integrating
the length of shortest paths and the volume element is induced by the
metric.This lead to the study of Metric Measured Spaces.

Motivated by optimal transport, Felix Otto and Cédric Villani made evident
the importance of the geometry of spaces of measures. Their efforts met
earlier ones made by probabilists such as Dominique Bakry and Michel Émery
who looked for conditions ensuring the validity of log-Sobolev inequalities.
Maxim Kontsevitch and John Lott rediscovered some of these facts in their
attempts to generalize the notion of Ricci curvature to non smooth metrics.

This very rich circle of ideas has become a very active area of research
allowing to revisit some classical domains of Analysis, such as solving
Monge Ampère equations. The intricate mixing of ideas coming from diverse
fields makes it quite remarkable.