  We address the well-posedness of the Muskat problem in a periodic geometry
and in a setting which allows us to consider general initial and boundary data,
gravity effects, as well as surface tension effects. In the absence of surface
tension we prove that the Rayleigh-Taylor condition identifies a domain of
parabolicity for the Muskat problem. This property is used to establish the
well-posedness of the problem. In the presence of surface tension effects the
Muskat problem is of parabolic type for general initial and boundary data. As a
bi-product of our analysis we obtain that Dirichlet-Neumann type operators
associated with certain diffraction problems are negative generators of
strongly continuous and analytic semigroups in the scale of small H\"older
spaces.
