The Muskat equation governs the motion of an interface e.g. between oil and water in sand. The equation is linearly stable as long as the oil stays on top of the water. The minicourse presents joint work of A. Castro, D. Cordoba, C. Fefferman and F. Gancedo. We show that the interface can "turn over" in finite time, i.e. somewhere the water will be on top of the oil. After a turnover, the solution is highly unstable but does not become singular immediately. However, a singularity can form later.