This book presents contributions of participants of a workshop held at the Centre de Recherches Mathematiques (CRM), University of Montreal. It can be viewed as a sequel to ""Mirror Symmetry I"" (1998), ""Mirror Symmetry II"" (1996), and ""Mirror Symmetry III"" (1999), copublished by the AMS and International Press. The volume presents a broad survey of many of the noteworthy developments that have taken place in string theory, geometry, and duality since the mid 1990s.Some of the topics emphasized include the following: integrable models and supersymmetric gauge theories; theory of M- and D-branes and noncommutative geometry; duality between strings and gauge theories; and, elliptic genera and automorphic forms. Several introductory articles present an overview of the geometric and physical aspects of mirror symmetry and of corresponding developments in symplectic geometry. The book provides an efficient way for a very broad audience of mathematicians and physicists to explore the frontiers of research into this rapidly expanding area.