  In this survey we present the state of the art about the asymptotic behavior
and stability of the modified Mullins--Sekerka flow and the surface diffusion
flow of smooth sets, mainly due to E.~Acerbi, N.~Fusco, V.Julin and M.Morini.
First we discuss in detail the properties of the nonlocal Area functional under
a volume constraint, of which the two flows are the gradient flow with respect
to suitable norms, in particular, we define the strict stability property for a
critical set of such functional and we show that it is a necessary and
sufficient condition for minimality under $W^{2,p}$-perturbations, holding in
any dimension. Then, we show that, in dimensions two and three, for initial
sets sufficiently "close" to a smooth strictly stable critical set $E$, both
flows exist for all positive times and asymptotically "converge" to a translate
of $E$.
