Off-shell (4, 0) supermultiplets in 2-dimensions are formulated. These are used
to construct sigma models whose target spaces are vector bundles over manifolds that are
hyperk¨ahler with torsion. The off-shell supersymmetry implies that the complex structures
are simultaneously integrable and allows us to write actions using extended superspace and
projective superspace, giving an explicit construction of the target space geometries.