  A general study of non-abelian duality is presented. We first identify a
possible obstruction to the conformal invariance of the dual theory for
non-semisimple groups. We construct the exact non-abelian dual for any
Wess-Zumino-Witten (WZW) model for any anomaly free subgroup, and the
corresponding extension for coset conformal field theories. We characterize the
exact non-abelian dual for $\sigma$-models with chiral isometries and extend
the standard notion of duality to anomalous subgroups of
  WZW models, thus giving a way of constructing dual transformations for
different groups on the left and on the right. We also present some new
examples of non-abelian duality for four-dimensional gravitational instantons.
