This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the topological recursion, a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall how computing
large size asymptotics in random matrices, has allowed to discover some fascinating and ubiquitous geometric invariants. Specializations of this method recover many classical invariants, like Gromov--Witten invariants, or knot polynomials (Jones, HOMFLY,...). In this short review, we give some examples, give definitions, and review some properties and applications of the formalism.
We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and
the correlation functions of the lattice) obey the topological recursion, as usual in matrix models, i.e they are given by the symplectic invariants of their spectral curve.
We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in [Eynard-Orantin]. As an application, we prove the
conjecture proposed by Bouchard and Marino, relating Hurwitz numbers to the spectral invariants of the Lambert curve exp(x)=y exp(-y).