For a bounded linear operator T on a complex Hilbert space and n ∈ N, T is said to be n-normal if T * T n = T n T *. In this paper we show that if T is a 2-normal operator and satisfies σ(T) ∩ (−σ(T)) ⊂ {0}, then T is isoloid and σ(T) = σ a (T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T − z) ⊥ ker(T − w). And if non-zero number z ∈ C is an isolated point of σ(T), then we show that ker(T − z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying σ(T) ∩ (−σ(T)) = ∅, then Weyl's theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n, m)-normal operators and show some properties of this kind of operators.