  We explore some of the global aspects of duality transformations in String
Theory and Field Theory. We analyze in some detail the equivalence of dual
models corresponding to different topologies at the level of the partition
function and in terms of the operator correspondence for abelian duality. We
analyze the behavior of the cosmological constant under these transformations.
We also explore several examples of non-abelian duality where the classical
background interpretation can be maintained for the original and the dual
theories. In particular we construct a non-abelian dual of $SL(2,R)$ which
turns out to be a three-dimensional black hole
