  A geometry with parallel skew-symmetric torsion is a Riemannian manifold
carrying a metric connection with parallel skew-symmetric torsion. Besides the
trivial case of the Levi-Civita connection, geometries with non-vanishing
parallel skew-symmetric torsion arise naturally in several geometric contexts,
e.g. on naturally reductive homogeneous spaces, nearly K\"ahler or nearly
parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or
twistor spaces over quaternion-K\"ahler manifolds with positive scalar
curvature. In this paper we study the local structure of Riemannian manifolds
carrying a metric connection with parallel skew-symmetric torsion. On every
such manifold one can define a natural splitting of the tangent bundle which
gives rise to a Riemannian submersion over a geometry with parallel
skew-symmetric torsion of smaller dimension endowed with some extra structure.
We show how previously known examples of geometries with parallel
skew-symmetric torsion fit into this pattern, and construct several new
examples. In the particular case where the above Riemannian submersion has the
structure of a principal bundle, we give the complete local classification of
the corresponding geometries with parallel skew-symmetric torsion.
