Common density functional approximations (DFAs) for the exchange-correlation energy suffer from self-interaction error (SIE), which is believed to be the cause of many of the failures of these approximations, such as poor description of charge transfer and transition states of chemical reactions. The standard self-interaction correction (SIC) of Perdew and Zunger (PZ) removes spurious self-interaction terms orbital-by-orbital. We implemented the Perdew–Zunger SIC self-consistently and carried out systematic tests of its performance. We found that PZ-SIC impairs the accuracy of semi-local functionals for equilibrium properties. PZ-SIC seems to overcorrect many-electron systems. We devised a modified version of the SIC, which is scaled down in many-electron regions. The scaled-down SIC has greatly improved performance for many molecular properties. Studies of fractionally-charged systems led to the new definition of "many-electron self-interaction error", which is a generalization of the one-electron concept. An " M -electron self-interaction-free" functional is one that produces a realistic linear variation of total energy with electron number N between the integers M -1 and M . Semi-local DFAs exhibit large many-electron SIE and therefore fail for systems with fractional average electron number. PZ-SIC and its scaled-down variants are one-electron SIE-free. PZ-SIC is often nearly many-electron SIE-free, but this property is lost in the scaled-down SIC. Another consequence of the SIE is incorrect asymptotic behavior of the exchange-correlation potential in semi-local DFAs. PZ-SIC recovers the exact asymptote, but its scaled-down version does not. An efficient method to impose the exact asymptote in a hybrid functional is to introduce range separation into the exchange component and replace the long-range portion of the approximate exchange by the Hartree-Fock counterpart. We show that this long-range correction works particularly well in combination with the short-range variant of the Perdew, Burke, and Ernzerhof (PBE) exchange functional. This long-range-corrected hybrid, denoted LC-ωPBE, is remarkably accurate for a broad range of molecular properties, such as thermo-chemistry, barrier heights of chemical reactions, bond lengths, and most notably, description of processes involving long-range charge transfer. Although LC-ωPBE is not exactly one-electron SIE-free, it can be nearly many-electron SIE-free in many cases.