A substantial fraction of prompt GRB spectra have soft spectral indexes exceeding the maximum allowed by the synchrotron model &agr;max= −2/3. Some spectra also exhibit a very sharp break atEp, inconsistent with the smooth synchrotron spectra. These facts pose a serious problem for the “optically thin synchrotron” interpretation of the prompt emission. We review various models suggested in order to resolve this puzzle. © 2004 American Institute of Physics