T - Duality and the Weakly Coupled Heterotic String

T-duality is a symmetry of the heterotic string to all orders in string perturbation theory. This results in an effective four dimensional supergravity theory with desirable features for phenomenology. T-duality, as well as, generically, an anomalous U(1), is broken by quantum anomalies of the effective field theory. The structure of the full anomaly is presented, and the mechanisms for anomaly cancellation are described. Plenary talk at SUSY09, June 10-25, 2009, Northeastern University; to be published in the proceedings. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract DE-AC02-05CH11231, in part by the National Science Foundation under grants PHY-0457315 and PHY99-07949.