Strings, higher curvature corrections, and black hole entropy

We review old and recent results about the correspondence between the states of black holes and string states. Due to quantum corrections the entropy of extremal black holes is corrected in two ways: first, there is an explicit modification of the geometry due to higher curvature terms in the effective action. Second, the area law itself is modified, as was first realized by R. Wald. When taking both kinds of corrections into account, it can be shown that not only the leading term but also subleading corrections to the black hole entropy match with the asymptotic state degeneracy predicted by string theory. Recently, H. Ooguri, A. Strominger and C. Vafa used this result to propose a non-perturbative definition of topological string theory. This bold idea has catalysed a renewed interest in the role of higher curvature corrections, and leads to several interesting open questions.