Classical continuum descriptions of material degradation may cease to be mathematically meaningful in case of softening-induced localisation of deformation. Several enhancements of conventional models have been proposed to remove this deficiency. The properties of two of these so-called regularisation methods, the nonlocal and the gradient approaches, are examined and compared in a continuum damage context. It is shown that the enhanced models allow for the propagation of waves in the softening zone, in contrast to conventional damage models. For both types of enhancement wave propagation becomes dispersive. The behaviour under quasi-static loading conditions is studied numerically. Finite element simulations of a one-dimensional problem yield quite similar results for the nonlocal and a gradient-enhanced model. The gradient enhancement has been used to model concrete fracture, yielding results which are in good agreement with experimental data.