  We describe how the dynamical system of rolling two $n$-dimensional
connected, oriented Riemannian manifolds $M$ and $\hat M$ without twisting or
slipping, can be lifted to a nonholonomic system of elements in the product of
the oriented orthonormal frame bundles belonging to the manifolds. By
considering the lifted problem and using properties of the elements in the
respective principal Ehresmann connections, we obtain sufficient conditions for
the local controllability of the system in terms of the curvature tensors and
the sectional curvatures of the manifolds involved. We also give some results
for the particular cases when $M$ and $\hat M$ are locally symmetric or
complete.
