If (M,g) and (M^,g^) are two smooth connected complete oriented Riemannian manifolds of dimensions n and n^ respectively, we model the rolling of (M,g) onto (M^,g^) as a driftless control affine systems describing two possible constraints of motion: the first rolling motion (Σ) NS captures the no-spinning condition only and the second rolling motion (Σ) R corresponds to rolling without spinning nor slipping. Two distributions of dimensions (n+n^) and n are then associated to the rolling motions (Σ) NS and (Σ) R respectively. This generalizes the rolling problems considered in Chitour and Kokkonen (Rolling manifolds and controllability: the 3D case, 2012) where both manifolds had the same dimension. The controllability issue is then addressed for both (Σ) NS and (Σ) R and completely solved for (Σ) NS . As regards to (Σ) R , basic properties for the reachable sets are provided as well as the complete study of the case (n,n^)=(3,2) and some sufficient conditions for non-controllability.