The Muskat problem models the interface dynamics between two incompressible, immiscible fluids of different physical characteristics in porous media. In particular, when the interface is close to the flat stable solution, one can study the regularity and large time behavior of the system. This talk will discuss recent results in the regime where the fluids are of different viscosities and different densities, including global existence and uniqueness in the critical space $\mathcal{F}^{1,1} \cap L^{2}$, instant analyticity of solutions, large time decay of the interface, gain of regularity in Sobolev spaces and ill-posedness in the unstable regime for low regularity solutions.