This article is a short review on the relationship between convergent matrix integrals, formal matrix integrals, and combinatorics of maps. studied for the eigenvalues statistical properties [48, 33, 5, 52] . Statistical properties of the spectrum of large random matrices show some amazing universal behaviours, and it is believed that they correspond to some kind of "central limit theorem" for non independent random variables. This domain is very active and rich, and many important recent

more »

... sses have been achieved by the mathematicians community. Universality was proved in many cases, in particular using the Riemann-Hilbert approach of Bleher-Its [6] and Deift Venakides Zhou Mac Laughlin [19] , and also by large deviation methods [34, 35] . -in Quantum Chromodynamics, quantum gravity, string theory, conformal field theory, formal matrix integrals are studied for their combinatorial property of being generating functions of maps [20] . This fact was first discovered by t 'Hooft in 1974 [49], then further developed mostly by BIPZ [12] as well as Ambjorn, David, Kazakov [20, 18, 32, 37, 38] . For a long time, physicist's papers have been ambiguous about the relationship between formal and convergent matrix integrals, and many people have been confused by those ill-defined matrix integrals. However, if one uses the word "formal matrix integral", many physicist's results of the 80's till now are perfectly rigorous, especially those using loop equations. Only results regarding convergency properties were non rigorous, but as far as combinatorics is concerned, convergency is not an issue. The ambiguity in physicist's ill-defined matrix integrals started to become obvious when E. Kanzieper and V. Freilikher [42], and later Brezin and Deo in 1998 [11] tried to compare the topological expansion of a formal matrix integral derived from loop equations, and the asymptotics of the convergent integral found with the help of orthogonal polynomials. The two results did not match. The orthogonal polynomial's method showed clearly that the convergent matrix integrals had no large N power series expansion (it contained some (−1) N ). The origin of this puzzle has now been understood [9] , and it comes from the fact that formal matrix integrals and convergent matrix integrals are different objects in general. This short review is only about combinatoric properties of formal matrix integrals. Matrix models is a very vast topic, and many important applications, methods and points of view are not discussed here. In particular, critical limits (which include asymptotics of combinatoric properties of maps), the link with integrable systems, with conformal field theory, with algebraic geometry, with orthogonal polynomials, group theory, number theory, probabilities and many other aspects, are far beyond the scope of such a short review.