  We prove the following quantitative version of the celebrated Soap Bubble
Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of
$\mathbb{R}^{n+1}$, $n\geq1$, and denote by $osc(H)$ the oscillation of its
mean curvature. We prove that there exists a positive $\varepsilon$, depending
on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that
if $osc(H) \leq \varepsilon$ then there exist two concentric balls $B_{r_i}$
and $B_{r_e}$ such that $S \subset \overline{B}_{r_e} \setminus B_{r_i}$ and
$r_e -r_i \leq C \, osc(H)$, with $C$ depending only on $n$ and upper bounds on
the surface area of $S$ and the $C^2$ regularity of $S$. Our approach is based
on a quantitative study of the method of moving planes and the quantitative
estimate on $r_e-r_i$ we obtain is optimal.
  As a consequence of this theorem, we also prove that if $osc(H)$ is small
then $S$ is diffeomorphic to a sphere and give a quantitative bound which
implies that $S$ is $C^1$-close to a sphere.
