  The goal of this "Habilitation \`a diriger des recherches" is to present two
different applications, namely computations of certain partition functions in
probability and applications to integrable systems, of the topological
recursion developed by B. Eynard and N. Orantin in 2007. Since its creation,
the range of applications of the topological recursion has been growing and
many results in different fields have been obtained. The first aspect that I
will develop deals with the historical domain of the topological recursion:
random matrix integrals. I will review the formalism of the topological
recursion 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 of
the topological recursion is that it can recover from the leading order of the
asymptotic all sub-leading orders with elementary computations. This method is
particularly 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 integrals
with hard edges, integrals over unitary matrices and much more. In the end, I
will also briefly mention the generalization to $\beta$-ensembles. In a second
chapter, I will review the connection between the topological recursion and the
study of integrable systems having a Lax pair representation. Most of the
results presented there will be illustrated by the case of the famous six
Painlev\'e equations. Though the formalism used in this chapter may look
completely disconnected from the previous one, it is well known that the local
statistics of eigenvalues in random matrix theory exhibit a universality
phenomenon and that the encountered universal systems are precisely driven by
some solutions of the Painlev\'{e} equations. As I will show, the connection
can be made very explicit with the topological recursion formalism.
