The Penrose inequality gives a lower bound for the total mass of a spacetime in terms of the area of suitable surfaces that represent black holes. Its validity is supported by the cosmic censorship conjecture and therefore its proof (or disproof) is an important problem in relation with gravitational collapse. The Penrose inequality is still a very challenging open problem in mathematical relativity, and it has received continuous attention since its formulation by Penrose in the early seventies. This talk will give a very brief and non-mathematical introduction on this topic. Some recent progresses and new interesting relative problems will also be mentioned.