We present the results of a comprehensive study of the gamma-ray burst 080928 and of its afterglow. GRB 080928 was a long burst detected by Swift/BAT and Fermi/GBM. It is one of the exceptional cases where optical emission had already been detected when the GRB itself was still radiating in the gamma-ray band. For nearly 100 s simultaneous optical, X-ray and gamma-ray data provide a coverage of the spectral energy distribution of the transient source from about 1 eV to 150 keV. In particular, we show that the SED during the main prompt emission phase agrees with synchrotron radiation. We constructed the optical/near-infrared light curve and the spectral energy distribution based on Swift/UVOT, ROTSE-IIIa (Australia), and GROND (La Silla) data and compared it to the X-ray light curve retrieved from the Swift/XRT repository. We show that its bumpy shape can be modeled by multiple energy-injections into the forward shock. Furthermore, weinvestigate whether the temporal and spectral evolution of the tail emission of the first strong flare seen in the early X-ray light curve can be explained by large-angle emission (LAE). We find that a nonstandard LAE model is required to explain the observations. Finally, we report on the results of our search for the GRB host galaxy, for which only a deep upper limit can be provided. Appendix A is only available in electronic form at http://www.aanda.org