New exchange-correlation functionals that address the important issue of many-electron self-interaction are developed. This is carried out by considering the performance of the functional on systems with fractional numbers of electrons at the same time as more standard thermochemical tests. The inclusion of Coulomb-attenuated exchange in the functional is facilitated by use of the adiabatic connection coupled with a short-range and long-range splittings. The new functionals have a good performance on thermochemistry and a much improved description of the total energy versus number of electrons and henceforth a much smaller many-electron self-interaction error.