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It was recently discovered (2004) that the coefficients F_g in the large N expansion of matrix integrals ln Z = Σ_g N ^2−2g F_g , obey a certain recursion relation. This recursion, and the coefficients F_g given by it, can also be defined by themselves, independently of any possibly underlying matrix model, they depend solely on a plane curve, called the spectral curve. The coefficients F_g have many properties, they are invariant under symplectic transformations of the spectral curve, their deformations satisfy some special geometry, their limits commute with resolving singularities, they are almost modular forms (and can be easily turned into modular forms), and their sum is the τ-function of an integrable system. They also have many applications, not only in random matrices. They count discrete surfaces (maps) of given genus, they count partitions and plane partitions, they count volumes of moduli spaces of Riemann surfaces (Mirzakhani's relations appear as a special case where the spectral curve is chosen to be y = sin√x ), and they are conjectured to give the Gromov-Witten invariants of every toric Calabi-Yau 3-fold when the spectral curve is the mirror curve of the Calabi-Yau manifold.