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We show theoretically that localized surface plasmons can generate optical fields with a chirality exceeding that of circularly polarized light by a factor of 50. This superchiral optical field can be formed from linearly polarized light incident on nonchiral metal structures. We identify three mechanisms that lead to large optical chirality involving the coupling between the incident light and the evanescent fields of the surface plasmons. Two of these mechanisms create superchiral regions with nonzero average chirality suitable for the excitation of chiral molecules in solution.