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