Exact Duality Symmetries in CFT and String Theory

The duality symmetries of WZW and coset models are discussed. The exact underlying symmetry responsible for semiclassical duality is identified with the symmetry under affine Weyl transformations. This identification unifies the treatement of duality symmetries and shows that in the compact and unitary case they are exact symmetries of string theory to all orders in $\alpha'$ and in the string coupling constant. Non-compact WZW models and cosets are also discussed. A toy model is analyzed suggesting that duality will not generically be a symmetry.