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This book is concerned with the classification of low dimensional nilpotent Leibniz algebras by central extensions over complex numbers. Leibniz algebras introduced by J.-L. Loday (1993) are non- antisymmetric generalizations of Lie algebras. There is a cohomology theory for these algebraic objects whose properties are similar to those of the classical Chevalley-Eilenberg cohomology theory for Lie algebras. The central extensions of Lie algebras play a central role in the classification theory of Lie algebras.