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In this survey we present the state of the art about the asymptotic behaviorand stability of the modified Mullins--Sekerka flow and the surface diffusionflow 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 undera volume constraint, of which the two flows are the gradient flow with respectto suitable norms, in particular, we define the strict stability property for acritical set of such functional and we show that it is a necessary andsufficient condition for minimality under $W^{2,p}$-perturbations, holding inany dimension. Then, we show that, in dimensions two and three, for initialsets sufficiently "close" to a smooth strictly stable critical set $E$, bothflows exist for all positive times and asymptotically "converge" to a translateof $E$.

Global Existence and Stability for the Modified Mullins-Sekerka and Surface Diffusion Flow