The overall performance of a Compton spectrometer and, in particular, its energy resolution are investigated both experimentally and theoretically for different scattering materials. Using low-Z (less than or equal to 8) scatterers of moderate sizes (scatterer diameter d less than or equal to 5 mm), there are negligible disturbances due to coherent and/or multiple scattering at 90 degrees scattering angle and photon energies above 20 keV. Two factors contribute to decreasing the energy resolution compared with that in direct measurements: (i) the velocity distribution of the electrons in the scatterer and (ii) the scattering geometry. Of these, (i) is dominant for photon energies less than or equal to 100 keV. The optimal scattering material is a metal of as low Z as possible, i.e. beryllium. However, polyethylene and lucite are normally sufficiently good scatterers. The scattering geometry may become the dominating factor decreasing energy resolution at high photon energies hv greater than or equal to 150 keV.