A class of two-dimensional globally scale-invariant, but not conformal
ly invariant, theories is obtained. These systems are identified in th
e process of discussing global and local scaling properties of models
related by duality transformations, based on non-semisimple isometry g
roups. The construction of the dual partner of a given model is follow
ed through; non-local as well as local versions of the former are disc
ussed.