B model Gromov Witten theory Mirror symmetry constructions The B model approach to topological string theory on Calabi Yau n folds Geometric quantizations with applications to Gromov Witten theory Some classical quantum aspects of Calabi Yau moduli The total ancestor potential in singularity theory Lecture notes on bihamiltonian structures and their central invariants This book collects various perspectives contributed by both mathematicians and physicists on the B model and its role in mirror symmetry. Mirror symmetry is an active topic of research in both the mathematics and physics communities but among mathematicians the “A model” half of the story remains much better understood than the B model. This book aims to address that imbalance. It begins with an overview of several methods by which mirrors have been constructed and from there gives a thorough account of the “BCOV” B model theory from a physical perspective; this includes the appearance of such phenomena as the holomorphic anomaly equation and connections to number theory via modularity. Following a mathematical exposition of the subject of quantization the remainder of the book is devoted to the B model from a mathematician’s point of view including such topics as polyvector fields and primitive forms Givental’s ancestor potential and integrable systems