  We present an intrinsic formulation of the kinematic problem of two
$n-$dimensional manifolds rolling one on another without twisting or slipping.
We determine the configuration space of the system, which is an
$\frac{n(n+3)}2-$dimensional manifold. The conditions of no-twisting and
no-slipping are decoded by means of a distribution of rank $n$. We compare the
intrinsic point of view versus the extrinsic one. We also show that the
kinematic system of rolling the $n$-dimensional sphere over $\mathbb R^n$ is
controllable. In contrast with this, we show that in the case of $SE(3)$
rolling over $\mathfrak{se}(3)$ the system is not controllable, since the
configuration space of dimension 27 is foliated by submanifolds of dimension
12.
