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The goal of this "Habilitation \`a diriger des recherches" is to present twodifferent applications, namely computations of certain partition functions inprobability and applications to integrable systems, of the topologicalrecursion developed by B. Eynard and N. Orantin in 2007. Since its creation,the range of applications of the topological recursion has been growing andmany results in different fields have been obtained. The first aspect that Iwill develop deals with the historical domain of the topological recursion:random matrix integrals. I will review the formalism of the topologicalrecursion as well as how it can be used to obtain asymptotic $\frac{1}{N}$series expansion of various matrix integrals. In particular, a key feature ofthe topological recursion is that it can recover from the leading order of theasymptotic all sub-leading orders with elementary computations. This method isparticularly well known and fruitful in the case of hermitian matrix integrals,but I will also show that the general method can be used to cover integralswith hard edges, integrals over unitary matrices and much more. In the end, Iwill also briefly mention the generalization to $\beta$-ensembles. In a secondchapter, I will review the connection between the topological recursion and thestudy of integrable systems having a Lax pair representation. Most of theresults presented there will be illustrated by the case of the famous sixPainlev\'e equations. Though the formalism used in this chapter may lookcompletely disconnected from the previous one, it is well known that the localstatistics of eigenvalues in random matrix theory exhibit a universalityphenomenon and that the encountered universal systems are precisely driven bysome solutions of the Painlev\'{e} equations. As I will show, the connectioncan be made very explicit with the topological recursion formalism.

Application de la r\'ecurrence topologique aux int\'egrales de matrices al\'eatoires et aux syst\`emes int\'egrables