  In this paper we consider the Muskat problem describing the motion of two
unbounded immiscible fluid layers with equal viscosities in vertical or
horizontal two-dimensional geometries. We first prove that the mathematical
model can be formulated as an evolution problem for the sharp interface
separating the two fluids, which turns out to be, in a suitable functional
analytic setting, quasilinear and of parabolic type. Based upon these
properties, we then establish the local well-posedness of the problem for
arbitrary large initial data and show that the solutions become instantly
real-analytic in time and space. Our method allows us to choose the initial
data in the class $H^s,$ $s\in(3/2,2)$, when neglecting surface tension,
respectively in $H^s,$ $s\in(2,3),$ when surface tension effects are included.
Besides, we provide new criteria for the global existence of solutions.
