Of concern is the moving boundary problem of a two-phase potential flow of two fluids with possible different densities and viscosities. Such problems are known as Muskat problems or two-phase Hele-Shaw flows. Due to the moving interfaces these problems are intrinsically nonlocal and highly nonlinear. A criterion is presented, known as the generalised Rayleigh–Taylor condition, which guarantees that for large classes of initial data these problems are classically well-posed, possibly on a finite time interval only. Away from this Rayleigh–Taylor regime the system becomes unstable and finger shaped unstable steady states can occur. A thin film approximation is also discussed. Here the dynamical behaviour is different: global weak solutions exist for any square integrable non-negative initial configuration. In addition, the flat steady state is globally stable in the class of weak solutions.