  Non-Abelian duality transformations built on non-semi-simple isometry groups
are analysed. We first give the conditions under which the original non-linear
sigma model and its non-Abelian dual are equivalent. The existence of an
invariant and non-degenerate bilinear form for the isometry Lie algebra is
crucial for this equivalence. The non-Abelian dual of a conformally invariant
sigma model, with non-semi-simple isometries, is then constructed and its beta
functions are shown to vanish. This study resolves an apparent obstruction to
the conformal invariance of sigma models obtained via non-Abelian duality based
on non-semi-simple groups.
