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Let H be a separable complex Hilbert space and B(H) is the algebra of all bounded linear operators on H. Normal operators on a Hilbert space H constitute the most important tractable class of operators known; the particular importance of normal operators is due to the spectral theorem. In this book new classes of operators which are generalizations of normal operators on a Hilbert space are introduced. Some of the basic properties with some examples are studied. Moreover, the product, direct sum, tensor product and the sum of finite numbers of these type are discussed and establish some inequalities of these operators. Furthermore, the behavior of the spectrum of them and its parts is tried to study. Finally, this book shows that there is a lot of work that can be done in the area of generalizations of normal operators on a Hilbert space.