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 twice the exponent of convergence of the associated Kleinian group. The proof of this theorem 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 the exponent of convergence.