Let T be a bounded linear operator on a complex Hilbert space and n, m ∈ N. Then T is said to be n-normal if T*Tn = TnT* and (n, m)-normal if T*mTn = TnT*m. In this paper, we study several properties of n-normal, (n, m)-normal operators. In particular, we prove that if T is 2-normal with σ(T) ⋂ (−σ(T)) ⊂ {0}, then T is polarloid. Moreover, we study subscalarity of n-normal operators. Also, we prove that if T is (n, m)-normal, then T is decomposable and Weyl's theorem holds for f (T), where f is an analytic function on σ(T) which is not constant on each of the components of its domain.