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Many-body corrections to neutrino scattering from electrons and nuclei are investigated. The Columb correlations of the electrons screen the weak vector neutral current. Electron screening is important only for electron neutrinos (or antineutrinos) and most effective for neutrino energies of several MeV or less. At electron densities relevant for supernovae, the total quasielastic nu-e cross section can be reduced by 50 percent, and elastic neutrino-nucleus scattering by up to 85 percent. (Author)