Rigidly and locally supersymmetric two-dimensional nonlinear σ-models with torsion

We give a general discussion of supersymmetric nonlinear sigma models with torsion. In the case of rigid supersymmetry, our results include models with nonvanishing Nijenhuis tensor. For local supersymmetry we limit the discussion to models with local scale invariance, for which we prove that the number of supersymmetries of a given chirality is at most four. For (4,4) local supersymmetry we find that torsion and quaternionic geometry exclude each other. (orig.)