  The target space of a (4,0) supersymmetric two-dimensional sigma model with
Wess-Zumino term has a connection with totally skew-symmetric torsion and
holonomy contained in Sp(n).Sp(1), QKT-connection. We study the geometry of
QKT-connections. We find conditions to the existence of a QKT-connection and
prove that if it exists it is unique. Studying conformal transformations we
obtain a lot of (compact) examples of QKT manifolds. We present a (local)
description of 4-dimensional homogeneous QKT structures relying on the known
result of naturally reductive homogeneous Riemannian manifolds. We consider
Einstein-like QKT manifold and find closed relations with Einstein-Weyl
geometry in dimension four.
