  Alexandrov's soap bubble theorem asserts that spheres are the only connected
closed embedded hypersurfaces in the Euclidean space with constant mean
curvature. The theorem can be extended to space forms and it holds for more
general functions of the principal curvatures.
  In this short review, we discuss quantitative stability results regarding
Alexandrov's theorem which have been obtained by the author in recent years. In
particular, we consider hypersurfaces having mean curvature close to a constant
and we quantitatively describe the proximity to a single sphere or to a
collection of tangent spheres in terms of the oscillation of the mean
curvature. Moreover, we also consider the problem in a non local setting, and
we show that the non local effect gives a stronger rigidity to the problem and
prevents the appearance of bubbling.
