Computing Z_top

This is the text of my habilitation thesis defended at the \'Ecole Normale Sup\'erieure. The topological string presents an arena in which many features of string theory proper, such as the interplay between worldsheet and target space descriptions or open-closed duality, can be distilled into computational techniques which yield results beyond perturbation theory. In this thesis, I will summarize my research activity in this area. The presentation is organized around computations of the topological string partition function Z_top based on various perspectives on the topological string.