Research in string theory has generated a rich interaction with algebraic geometry, with exciting new work that includes the Strominger-Yau-Zaslow conjecture. This monograph builds on lectures at the 2002 Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string theory and algebraic geometry, presenting an updated discussion that includes subsequent developments. The group of distinguished mathematicians and mathematical physicists who produced this monograph worked as a team to create a unique volume. Its overall goal is to explore the physical and mathematical aspects of Dirichlet branes. The narrative is organized around two principal ideas: Kontsevich's Homological Mirror, Symmetry conjecture, and the Strominger-Yau-Zaslow conjecture. The authors explain how Kontsevich's conjecture is equivalent to the identification of two different categories of Dirichlet branes. They also explore the ramifications and current state of the Strominger-Yau-Zaslow conjecture. They relate the ideas to active areas of research that include the McKay correspondence, topological quantum field theory, and stability structures. The authors were not satisfied to tell their story twice, from separate mathematics and physics points of view. Instead, theirs is a unified presentation offered in a way that both mathematicians and physicists can follow, without having all of the foundations of both subjects at their immediate disposal.