A strong geometry with torsion is a Riemannian manifold carrying a metric connection with closed skew-symmetric torsion. In the seminar I will first review general properties of metric connections with closed skew-symmetric torsion. Then I will focus on the case of Hermitian manifolds and $7$-manifolds endowed with a $G_2$-structure. This talk is part of the activities of the FIS2 project BeyondCY3 - CUP B53C25001070001. More information available at: https://sites.google.com/uniroma1.it/lfoscolo/fis2-beyondcy3/events