This paper addresses the issue of self-interaction error in density-functional approximations for many-electron systems. The exact density functional for the ground-state energy is self-interaction-free, but many approximations, including the local-spin-density (LSD) approximation for exchange and correlation, are not. The authors propose two methods for self-interaction correction (SIC) of any density functional for the energy. These methods are sanctioned by the Hohenberg-Kohn theorem and are based on the variational principle. The first method introduces an orbital-dependent single-particle potential, while the second involves a local potential as in the Kohn-Sham scheme. The authors apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole while substantially improving the description of its shape. They apply this method to various physical problems where the uncorrected LSD approach produces systematic errors. The results show systematic improvements, both qualitative and quantitative, from this simple correction. Benefits of SIC in atomic calculations include improved values for the total energy and for the separate exchange and correlation pieces of it, accurate binding energies of negative ions, more accurate electron densities, orbital eigenvalues that closely approximate physical removal energies, and correct long-range behavior of the potential and density. The authors also discuss the admissibility of fractional occupation numbers and present a parametrization of the electron-gas correlation energy at any density based on recent results of Ceperley and Alder. The paper also discusses the exchange-correlation hole and its importance in density-functional theory. The authors show that the LSD approximation gives reasonable results for realistic systems that are formally outside its domain of validity because it exactly satisfies the sum rule on the number content of the exchange-correlation hole. The SIC-LSD approximation improves upon this by correcting the self-interaction error. The authors also discuss the eigenvalues and removal energies in the context of density-functional theory. They show that the SIC-LSD approximation provides a more accurate description of the removal energies compared to the LSD approximation. The results demonstrate that the SIC-LSD approximation can significantly improve the accuracy of the calculated energies and other physical quantities in many-electron systems.This paper addresses the issue of self-interaction error in density-functional approximations for many-electron systems. The exact density functional for the ground-state energy is self-interaction-free, but many approximations, including the local-spin-density (LSD) approximation for exchange and correlation, are not. The authors propose two methods for self-interaction correction (SIC) of any density functional for the energy. These methods are sanctioned by the Hohenberg-Kohn theorem and are based on the variational principle. The first method introduces an orbital-dependent single-particle potential, while the second involves a local potential as in the Kohn-Sham scheme. The authors apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole while substantially improving the description of its shape. They apply this method to various physical problems where the uncorrected LSD approach produces systematic errors. The results show systematic improvements, both qualitative and quantitative, from this simple correction. Benefits of SIC in atomic calculations include improved values for the total energy and for the separate exchange and correlation pieces of it, accurate binding energies of negative ions, more accurate electron densities, orbital eigenvalues that closely approximate physical removal energies, and correct long-range behavior of the potential and density. The authors also discuss the admissibility of fractional occupation numbers and present a parametrization of the electron-gas correlation energy at any density based on recent results of Ceperley and Alder. The paper also discusses the exchange-correlation hole and its importance in density-functional theory. The authors show that the LSD approximation gives reasonable results for realistic systems that are formally outside its domain of validity because it exactly satisfies the sum rule on the number content of the exchange-correlation hole. The SIC-LSD approximation improves upon this by correcting the self-interaction error. The authors also discuss the eigenvalues and removal energies in the context of density-functional theory. They show that the SIC-LSD approximation provides a more accurate description of the removal energies compared to the LSD approximation. The results demonstrate that the SIC-LSD approximation can significantly improve the accuracy of the calculated energies and other physical quantities in many-electron systems.