The possibility that the diffuse interstellar lines and bands are but structure in the continuous extinction by dust grains is considered in detail. The lines are assumed to arise from impurities in the grains. Profiles of the strongest diffuse features are computed for a wide variety of host grain types and sizes, including size distributions of particles, and spheroidal and coated spherical grains. New observational profiles, of which ‘4430’, ‘4765’ and ‘4885’ have been derived from automated spectrophotometry of electronographic spectra, are also presented; and comparison is made between theory and observation. The most useful tool of comparison is found to be the ratio of maximum apparent emission in the violet wing to maximum apparent absorption, ϱ. This ratio increases with size of the host particle, and with both real and imaginary parts of its refractive index; but is independent of the observed variations in strength and width of a particular line. The ratio is larger for ‘4430’ than for the other lines. Taking into account current opinion as to the composition of interstellar dust, the observations may be explained by the presence of impurities in either silicate grains with effective radii of about 120 nm, or possibly graphite grains about 50 nm in radius coated with a thin dielectric condensate. None of the observed profiles are at variance with the hypothesis that all the lines arise from the same impurities in the same type of grains. The concentration of such impurities needed to reproduce the observed line profiles and strengths is only about one in 104 atoms in the grain.