  We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with
$\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from
below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the
classic L\'evy-Gromov isoperimetric inequality (together with the recent
sharpening counterparts proved in the smooth setting by E. Milman for any $K\in
\mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetric
profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the
isoperimetric profile of the model space. Moreover, if equality is attained for
some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be
a spherical suspension and in this case we completely classify the
isoperimetric regions. Finally we also establish the almost rigidity: if the
equality is almost attained for some volume $v \in (0,1)$ and $K$ is strictly
positive, then the space must be mGH close to a spherical suspension.
  To our knowledge this is the first result about isoperimetric comparison for
non smooth metric measure spaces satisfying Ricci curvature lower bounds.
Examples of spaces fitting our assumptions include measured Gromov-Hausdorff
limits of Riemannian manifolds satisfying Ricci curvature lower bounds and
Alexandrov spaces with curvature bounded from below; the result seems new even
in these celebrated classes of spaces.
