  On a compact complex manifold we study the behaviour of strong K\"ahler with
torsion (strong KT) structures under small deformations of the complex
structure and the problem of extension of a strong KT metric. In this context
we obtain the analogous result of Miyaoka extension theorem. Studying the
blow-up of a strong KT manifold at a point or along a complex submanifold, we
prove that a complex orbifold endowed with a strong KT metric admits a strong
KT resolution. In this way we obtain new examples of compact simply-connected
strong KT manifolds.
