  A geometric inequality in General Relativity relates quantities that have
both a physical interpretation and a geometrical definition. It is well known
that the parameters that characterize the Kerr-Newman black hole satisfy
several important geometric inequalities. Remarkably enough, some of these
inequalities also hold for dynamical black holes. This kind of inequalities
play an important role in the characterization of the gravitational collapse,
they are closed related with the cosmic censorship conjecture. Axially
symmetric black holes are the natural candidates to study these inequalities
because the quasi-local angular momentum is well defined for them. We review
recent results in this subject and we also describe the main ideas behind the
proofs. Finally, a list of relevant open problem is presented.
