The collection of articles in this volume are based on lectures presented during the Winter School on Mirror Symmetry held at Harvard University. There are many new directions suggested by mirror symmetry which could potentially have very rich connections in physics and mathematics. This book brings together the latest research in a major area of mathematical physics, including the recent progress in mirror manifolds and Lagrangian submanifolds. In particular, several articles describing homological approach and related topics are included. Other AMS titles edited by S.-T Yau published in the AMS/IP Studies in Advanced Mathematics series include, Mirror Symmetry III, Volume 10, Mirror symmetry II, Volume 1, and Mirror Symmetry I, Volume 9.