  We consider N=2 and N=4 supersymmetric gauge theories in two-dimensions,
coupled to matter multiplets. In analogy with the N=2 case also in the N=4 case
one can introduce Fayet-Iliopoulos terms.The associated three-parameters have
the meaning of momentum-map levels in a HyperK\"ahler quotient construction.
Differently from the N=2 case, however, the N=4 has a single phase
corresponding to an effective $\sigma$-model. There is no Landau-Ginzburg
phase. The main possible application of our N=4 model is to an effective
Lagrangian construction of a $\sigma$-model on ALE-manifolds. We discuss the A
and B topological twists of these models clarifying some issues not yet
discussed in the literature, in particular the identification of the
topological systems emerging from the twist. Applying our results to the case
of ALE-manifolds we indicate how one can use the topologically twisted theories
to study the K\"ahler class and complex structure deformations of these
gravitational instantons.
