Continuous models of material degradation may cease to producemeaningful results in the presence of high strain gradients. Thesegradients may occur for instance in the propagation of waves withhigh wave numbers and at stress concentrators. Adding nonlocalor gradient terms to the constitutive modelling may enhance theability of the models to describe such situations. The effect of addingnonlocal or gradient terms and the relation between theseenhancements are examined in a continuum damage setting. Anonlocal damage model and two different gradient damage modelsare considered. In one of the gradient models higher orderdeformation gradients enter the equilibrium equations explicitly, whilein the other model the gradient influence follows in a more implicitway from an additional partial differential eq! uation. The latter, implicitgradient formulation can be rewritten in the integral format of thenonlocal model and can therefore be regarded as truly nonlocal.This is not true for the explicit formulation, in which the nonlocality islimited to an infinitesimal volume. This fundamental differencebetween the formulations results in quite different behaviour in wavepropagation, localisation and at crack tips. This is shown for thepropagation of waves in the models, their localisation properties andthe behaviour at a crack tip. The responses of the nonlocal modeland the implicit gradient model agree remarkably well in thesesituations, while the explicit gradient formulation shows an entirelydifferent and sometimes nonphysical response.