A hypercomplex Hermitian metric is called HKT when the three associated Bismut connections coincide, yielding a unique connection with skew-symmetric torsion. This gives a natural generalization of the hyper-Kähler condition, where torsion is allowed to appear. The original definition additionally required this torsion to be closed, that is the strong HKT case, but this hypothesis was later dropped to include a broader family of examples. In this talk, I will first give an overview of the general HKT theory: basic definitions, classical examples, and their appearance in supersymmetric sigma models. I will then focus on the strong HKT setting, discussing recent results and rigidity phenomena on compact manifolds. I will conclude by outlining some open questions and mentioning ongoing joint work.
