We show that photoabsorption by fullerenes and buckyonions (multishell fullerenes) explain the shape, width, and peak energy of the most prominent feature of interstellar absorption, the UV bump at 2175 Å. The predicted optical and near-infrared transitions for these molecules also offer a potential explanation for the long-standing problem of the identity of the diffuse interstellar bands. The implied ubiquitous distribution of fullerenes may also account for the anomalous galactic microwave emission detected by cosmic microwave background experiments. Comparing theoretical cross sections and astronomical data, we estimate a density of fullerenes in the diffuse interstellar medium of 0.1-0.2 parts per million, consistent with the findings in meteorites. Fullerene-based molecules appear to be a major carbon reservoir in the interstellar medium.