These are the proceedings of the joint seminar by M.I.T. and Harvard on the current Developments in mathematics for the year 2001. Established in 1995, this seminar has been continued on the third weekend of November every year. The organizing committee for the seminar consisted of distinguished mathematicians from the mathematics departments of both institutions: Barry Mazur, Wilfried Schmid, and Shing-Tung Yau from Harvard, and David Jerison, A. J. de Jong and George Lustig from M.I.T.. We trust that these proceedings will be of interest to many mathematicians, and will inspire future developments and research pursuits in mathematics.