  This paper is devoted to a systematic study and classification of invariant
affine or metric connections on certain classes of naturally reductive spaces.
For any non-symmetric, effective, strongly isotropy irreducible homogeneous
Riemannian manifold $(M=G/K, g)$, we compute the dimensions of the spaces of
$G$-invariant affine and metric connections. For such manifolds we also
describe the space of invariant metric connections with skew-torsion. For the
compact Lie group ${\rm U}_{n}$ we classify all bi-invariant metric
connections, by introducing a new family of bi-invariant connections whose
torsion is of vectorial type. Next we present applications related with the
notion of $\nabla$-Einstein manifolds with skew-torsion. In particular, we
classify all such invariant structures on any non-symmetric strongly isotropy
irreducible homogeneous space.
