  The well known relation between extended supersymmetry and complex geometry
in the non-linear sigma-models is reviewed, and some recent developments
related to the introduction of the non-anti-commutativity, in the context of
the supersymmetric non-linear sigma-models formulated in extended superspace,
are discussed. This contribution is suitable for both physicists and
mathematicians interesting in the interplay between geometry, supersymmetry and
noncommutativity.
