According to semiempirical models, photoabsorption by fullerenes (single and multishell) could explain the shape, width and peak energy of the most prominent feature of the interstellar absorption, the UV bump at 2175 Å. Other weaker transitions are predicted in the optical and near-infrared providing a potential explanation for diffuse interstellar bands. In particular, we find that several fullerenes could contribute to the well known strong DIB at 4430 Å. Comparing cross sections and available data for this DIB and the UV bump we estimate a density of fullerenes in the diffuse interstellar medium of 0.1 0.2 ppm. These molecules could then be a major reservoir for interstellar carbon. We also study the rotation rates and electric dipole emission of hydrogenated icosahedral fullerenes. We investigate these molecules as potential carriers of the anomalous (dust-correlated) microwave emission recently detected by several cosmic microwave background experiments.