  We investigate the relation between supersymmetry and geometry for two
dimensional sigma models with target spaces of arbitrary signature, and
Lorentzian or Euclidean world-sheets. In particular, we consider twisted forms
of the two-dimensional $(p,q)$ supersymmetry algebra. Superspace formulations
of the $(p,q)$ heterotic sigma-models with twisted or untwisted supersymmetry
are given. For the twisted (2,1) and the pseudo-K\"{a}hler sigma models, we
give extended superspace formulations.
