  We sketch our recent application of a non-commutative version of the Cartan
`moving-frame' formalism to the quantum Euclidean space $R^N_q$, the space
which is covariant under the action of the quantum group $SO_q(N)$. For each of
the two covariant differential calculi over $R^N_q$ based on the $R$-matrix
formalism, we summarize our construction of a frame, the dual inner
derivations, a metric and two torsion-free almost metric compatible covariant
derivatives with a vanishing curvature. To obtain these results we have
developed a technique which fully exploits the quantum group covariance of
$R^N_q$. We first find a frame in the larger algebra $\Omega^*(R^N_q) \cocross
\uqs$. Then we define homomorphisms from $R^N_q \cocross U_q^{\pm}{so(N)}$ to
$R^N_q$ which we use to project this frame in $\Omega^*(R^N_q)$.
