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Complex analysis is a powerful tool to study classical integrable systems,statistical physics on the random lattice, random matrix theory, topologicalstring theory,... All these topics share certain relations, called "loopequations" or "Virasoro constraints". In the simplest case, the completesolution of those equations was found recently: it can be expressed in theframework of differential geometry over a certain Riemann surface which dependson the problem : the "spectral curve". This thesis is a contribution to thedevelopment of these techniques, and to their applications. Keywords: randommatrices, random maps, integrable systems, algebraic geometry, loop equations,topological recursion, Hurwitz numbers, Gromov-Witten invariants, Tracy-Widomlaws, beta ensemble, large N asymptotics in random matrix theory.

Quelques probl\`emes de g\'eom\'etrie \'enum\'erative, de matrices  al\'eatoires, d'int\'egrabilit\'e, \'etudi\'es via la g\'eometrie des  surfaces de Riemann