  Following the point of view of Gray and Hervella, we derive detailed
conditions which characterize each one of the classes of almost
quaternion-Hermitian $4n$-manifolds, $n>1$. Previously, by completing a basic
result of A. Swann, we give explicit descriptions of the tensors contained in
the space of covariant derivatives of the fundamental form $\Omega$ and split
the coderivative of $\Omega$ into its ${\sl Sp}(n){\sl Sp}(1)$-components. For
$4n>8$, A. Swann also proved that all the information about the intrinsic
torsion $\nabla \Omega$ is contained in the exterior derivative ${\sl d}
\Omega$. Thus, we give alternative conditions, expressed in terms of ${\sl d}
\Omega$, to characterize the different classes of almost quaternion-Hermitian
manifolds.
