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The 2-matrix model has been introduced to study Ising model on randomsurfaces. Since then, the link between matrix models and combinatorics ofdiscrete surfaces has strongly tightened. This manuscript aims to investigatethese deep links and extend them beyond the matrix models, following my work'sevolution. First, I take care to define properly the hermitian 2 matrix modelwhich gives rise to generating functions of discrete surfaces equipped with aspin structure. Then, I show how to compute all the terms in the topologicalexpansion of any observable by using algebraic geometry tools. They areobtained as differential forms on an algebraic curve associated to the model:the spectral curve. In a second part, I show how to define such differentialson any algebraic curve even if it does not come from a matrix model. I thenstudy their numerous symmetry properties under deformations of the algebraiccurve. In particular, I show that these objects coincide with the topologicalexpansion of the observable of a matrix model if the algebraic curve is thespectral curve of this model. Finally, I show that fine tuning the parametersensure that these objects can be promoted to modular invariants and satisfy theholomorphic anomaly equation of the Kodaira-Spencer theory. This gives a newhint that the Dijkgraaf-Vafa conjecture is correct.

From matrix models' topological expansion to topological string theories: counting surfaces with algebraic geometry