Geometry of hyper-Kahler connections with torsion

The internal space of a N = 4 supersymmetric model with Wess-Zumino term ha s a connection with totally skew-symmetric torsion and holonomy in Sp(n). W e study the mathematical background of this type of connection. In particul ar, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic dat a, develop its potential theory and reduction theory.