In this work we investigate the question of the well-posedness of the Muskat problem when low regularity initial data is considered. A natural barrier for well-posedness are the spaces that are critical under the scaling, and therefore an interesting question is if the well-posedness can be established for critical spaces and super-critical spaces. For Navier-Stokes this question was answered negatively in [2], [8], [23] and many other works since then for some other fluid equations, by showing that for some critical spaces the solution map is discontinuous at the origin. The first part of this work introduces the technical tools, approximations and explain the strategy that is used to prove the ill-posedness result for the Muskat equation. The next two chapters are dedicated to fill some gaps in the well-posedness theory for the Muskat problem by establishing global existence results for the 2D problem in a periodic domain. In Chapter 2 we prove global existence in a periodic domain for small initial data in the critical space $\mathcal{F}^{1,1}$, the analogous result was previously known for the non-periodic case in [10], [9]. In Chapter 3 we prove the global existence for $H^2$ initial data with small slope in a periodic domain by extending a result previously known for the non-periodic case [11]. The last part of the work is devoted to study the question of Ill-posedness for the Muskat equation and the Epitaxial Growth problem. We consider a family of approximations of the equation for which we prove the discontinuity of the solution map at the origin in some supercritical spaces. The sequence of spaces approaches a critical one as we consider higher order approximations which suggest that well-posedness in critical spaces is really the best we should hope for.