This book presents surveys from a workshop held during the theme year in geometry and topology at the Centre de recherches mathematiques (CRM, University of Montreal). The volume is in some sense a sequel to ""Mirror Symmetry I"" (1998) and ""Mirror Symmetry II"" (1996), copublished by the AMS and International Press. Included are recent developments in the theory of mirror manifolds and the related areas of complex and symplectic geometry. The long introductory articles explain the key physical ideas and motivation, namely conformal field theory, supersymmetry, and string theory. Open problems are emphasized.Thus the book provides an efficient way for a very broad audience of mathematicians and physicists to reach the frontier of research in this fast expanding area. It features: crucial research pertaining to future developments in algebraic and symplectic geometry and to the physics of unified string theories; well-known authors who are leaders in the field; introductory article by Greene and Yau; and a solid and even blend of ideas and techniques from both mathematics and physics. Also available from the AMS is the related volume, ""Mirror Symmetry IV"" (2002).