  A detailed study is made of the noncommutative geometry of $R^3_q$, the
quantum space covariant under the quantum group $SO_q(3)$. For each of its two
$SO_q(3)$-covariant differential calculi we find its metric, the corresponding
frame and two torsion-free covariant derivatives that are metric compatible up
to a conformal factor and which yield both a vanishing linear curvature. A
discussion is given of various ways of imposing reality conditions. The
delicate issue of the commutative limit is discussed at the formal algebraic
level. Two rather different ways of taking the limit are suggested, yielding
respectively $S^2\times R$ and $R^3$ as the limit Riemannian manifold.
