  The characteristic connection of an almost hermitian structure is a hermitian
connection with totally skew-symmetric torsion. The case of parallel torsion in
dimension six is of particular interest. In this work, we give a full
classification of the algebraic types of the torsion form, and, based on this,
undertake a systematic investigation into the possible geometries. Numerous
naturally reductive spaces are constructed and classified as well as examples
on nilmanifolds given.
