As an alternative to the standard Kohn-Sham procedure, other exact realizations of density-functional theory ~generalized Kohn-Sham methods! are presented. The corresponding generalized Kohn-Sham eigenvalue gaps are shown to incorporate part of the discontinuity D xc of the exchange-correlation potential of standard KohnSham theory. As an example, a generalized Kohn-Sham procedure splitting the exchange contribution to the total energy into a screened, nonlocal and a local density component is considered. This method leads to band gaps far better than those of local-density approximation and to good structural properties for the materials Si, Ge, GaAs, InP, and InSb.