$SU(2)\times SU(2)$ harmonic superspace and (4,4) sigma models with   torsion
  We review a manifestly supersymmetric off-shell formulation of a wide class of torsionful $(4,4)$ $2D$ sigma models and their massive deformations in the harmonic superspace with a double set of $SU(2)$ harmonic variables. Sigma models with both commuting and non-commuting left and right complex structures are treated. 
Evgeny A. Ivanov