Using the discontinuity of the chemical potential as a function of excess charge, fundamental gaps for finite systems and the band gaps of extended solids are determined within reduced density matrix functional theory. We also present the necessary and sufficient conditions for the one-body reduced density matrix of a system with fractional charge to be ensemble N-representable. The performance of most modern day reduced density matrix functionals is assessed for the gaps and the correlation energy of finite systems. Our results show that for finite systems the PNOF, BBC3, and power functionals yield very accurate correlation energies while for a correct description of the fundamental gap the removal of self-interaction terms is essential. For extended solids we find that the power functional captures the correct band gap behavior for conventional semiconductors as well as strongly correlated Mott insulators, where a gap is obtained in absence of any magnetic ordering.