  We review non-linear sigma-models with (2,1) and (2,2) supersymmetry. We
focus on off-shell closure of the supersymmetry algebra and give a complete
list of (2,2) superfields. We provide evidence to support the conjecture that
all N=(2,2) non-linear sigma-models can be described by these fields. This in
its turn leads to interesting consequences about the geometry of the target
manifolds. One immediate corollary of this conjecture is the existence of a
potential for hyper-Kahler manifolds, different from the Kahler potential,
which does not only allow for the computation of the metric, but of the three
fundamental two-forms as well. Several examples are provided: WZW models on
SU(2) x U(1) and SU(2) x SU(2) and four-dimensional special hyper-Kahler
manifolds.
