Résume | We review the world-sheet derivation of the holomorphic anomaly equations fulfilled by the all genus topological string partition function $Z$ on Calabi-Yau 3-folds $M$. Interpreting $Z$ as a wave function on $H_3(M, R)$ these equations can be viewed as describing infinitesimal changes of the symplectic frame. A recursive solution for $Z$ to high genus is provided using modular building blocks obtained by the periods of $M$ as well as constraints on the local expansion of $Z$ near singular loci in the complex moduli space of M in appropriate symplectic frames. Some recent applications of these ideas to elliptic fibred Calabi-Yau spaces are given. |