A systematic method which allows the optimum thickness of each layer in a depth-graded multilayer coating to be determined is described. This enables specific reflectivity responses over broad wavelength bands in the soft X-ray and EUV regions to be calculated. The method is applied to the design of some depth-graded molybdenumrsilicon multilayers for the wavelength range 13-19nm, with average normal incidence reflectivities of about 13% in this range, but it is generally applicable for other material pairs and wavelength ranges. In addition, the effects of layer thickness errors on the performance of depth-graded multilayers can be simulated. The model gives better results than those based on power law variation of the layer thicknesses.