On any Hermitian manifold there is a unique Hermitian connection, called the Bismut connection, which has torsion a three-form. One says that the triplet consisting of the Hermitian structure together with the Bismut connection specifies a Kähler-with-torsion structure, or briefly a KT structure. If the torsion three-form is closed, we have a strong KT structure. The first part of this talk will discuss these notions and also address the problem of classifying strong KT structures.
\paragraph{} Despite their name, KT manifolds are generally not Kähler. In particular the fundamental two-form is not closed. If the KT structure is strong, we have instead a closed three-form. Motivated by the usefulness of moment maps in geometries involving symplectic forms, one may ask whether it is possible to construct a similar type of map, when we replace the symplectic form by a closed three-form. The second part of the talk will explain the construction of such maps, which are called multi-moment maps.