  The scope of this note is to make a self-contained survey of the recent
developments and achievements of the theory of L1-Optimal Transportation on
metric measure spaces. Among the results proved in the recent papers [20, 21]
where the author, together with A. Mondino, proved a series of sharp (and in
some cases rigid) geometric and functional inequalities in the setting of
metric measure spaces enjoying a weak form of Ricci curvature lower bound, we
review the proof of the L\'evy-Gromov isoperimetric inequality.
