  We present a review of the main aspects of geometrothermodynamics, an
approach which allows us to associate a specific Riemannian structure to any
classical thermodynamic system. In the space of equilibrium states, we consider
a Legendre invariant metric, which is given in terms of the fundamental
equation of the corresponding thermodynamic system, and analyze its geometric
properties in the case of the van der Waals gas, and black holes. We conclude
that the geometry of this particular metric reproduces the thermodynamic
behavior of the van der Waals gas, and the Reissner-Nordstr\"om black hole, but
it is not adequate for the thermodynamic description of Kerr black holes.
