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This thesis deals with the geometric and integrable aspects associated withrandom matrix models. Its purpose is to provide various applications of randommatrix theory, from algebraic geometry to partial differential equations ofintegrable systems. The variety of these applications shows why matrix modelsare important from a mathematical point of view.  First, the thesis will focus on the study of the merging of two intervals ofthe eigenvalues density near a singular point. Specifically, we will show whythis special limit gives universal equations from the Painlev\'e II hierarchyof integrable systems theory. Then, following the approach of (bi) orthogonalpolynomials introduced by Mehta to compute partition functions, we will findRiemann-Hilbert and isomonodromic problems connected to matrix models, makingthe link with the theory of Jimbo, Miwa and Ueno. In particular, we willdescribe how the hermitian two-matrix models provide a degenerate case ofJimbo-Miwa-Ueno's theory that we will generalize in this context. Furthermore,the loop equations method, with its central notions of spectral curve andtopological expansion, will lead to the symplectic invariants of algebraicgeometry recently proposed by Eynard and Orantin. This last point will begeneralized to the case of non-hermitian matrix models (arbitrary $\beta$)paving the way to "quantum algebraic geometry" and to the generalization ofsymplectic invariants to "quantum curves". Finally, this set up will be appliedto combinatorics in the context of topological string theory, with the explicitcomputation of an hermitian random matrix model enumerating the Gromov-Witteninvariants of a toric Calabi-Yau threefold.

Aspects g\'eom\'etriques et int\'egrables des mod\`eles de matrices al\'eatoires