In this paper we study the set of bounded geodesics on a general, geometrically finite $(N+1)$-manifold of constant negative curvature. We obtain the result that the Hausdorff dimension of this set is equal to $2 \delta$, where $\delta$ denotes the exponent of convergence of the associated Kleinian group. The proof of this shows, in particular, that if the group has parabolic elements, then the set of limit points which are badly approximable with respect to the parabolic fixed points has Hausdorff dimension equal to $\delta$.