Non-anti-commutative deformation of complex geometry

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.