  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.
