In this paper we study discrepancy groups (d-groups), that are Kleinian groups whose exponent of convergence is strictly less than the Hausdorff dimension of their limit set. We show that the limit set of a d-group always contains continuous families of fractal sets, each of which contains the set of radial limit points and has Hausdorff dimension strictly less than the Hausdorff dimension of the whole limit set. Subsequently, we consider special d-groups which are normal subgroups of some geometrically finite Kleinian group. For these we obtain the result that their Poincare exponent is always bounded from below by half of the Poincare exponent of the associated geometrically finite group in which they are normal. Finally, we give a discussion of various examples of d-groups, which in particular also contains explicit constructions of these groups.