The Penrose inequality is a conjecture relating the total mass of asymptotically flat space-times satisfying appropriate curvature conditions and the area of certain closed, space-like and co-dimension-two surfaces describing black holes from a quasi-local perspective. This conjecture is motivated by physical properties of black holes and has been proven in a number of particular but very interesting cases, mainly concerning the so-called Riemannian Penrose inequality. In this talk I will present the general inequality, explain in which sense it is supported by black hole physics and describe several known results in the Riemannian case, including recent results for graphs in Euclidean space where the proof becomes particularly simple. If time permits, an approach to the inequality in the general case will also be mentioned.