  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 a very challenging problem in mathematical
relativity and it has received continuous attention since its formulation by
Penrose in the early seventies. Important breakthroughs have been made in the
last decade or so, with the complete resolution of the so-called Riemannian
Penrose inequality and a very interesting proposal to address the general case
by Bray and Khuri. In this paper, the most important results on this field will
be discussed and the main ideas behind their proofs will be summarized, with
the aim of presenting what is the status of our present knowledge in this
topic.
