In Peller (1985, 1990) [10,11], Aleksandrov and Peller (2009, 2010, 2010) [1-3] sharp estimates for f(A)-f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this Note we extend those results to the case of functions of normal operators. We show that if f belongs to the Hölder class Λ_α(R²), 0 < α < 1, of functions of two variables, and N1 and N2 are normal operators, then {norm of matrix}f(N₁)-f(N₂){norm of matrix} ≤ const{norm of matrix}f{norm of matrix}Λ_α{norm of matrix}N₁-N₂{norm of matrix}^α. We obtain a more general result for functions in the space Λ_ω(R²)={f:|f(ζ₁)-f (ζ₂)| ≤ const ω(|ζ₁-ζ₂|)} for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class B∞11(R2), then it is operator Lipschitz, i.e., {norm of matrix}f(N₁)-f(N₂){norm of matrix} ≤ const{norm of matrix}f{norm of matrix}B_∞1¹{norm of matrix}N₁-N₂{norm of matrix}. We also study properties of f(N₁)-f(N₂) in the case when f ∈ Λ α(R²) and N₁-N₂ belongs to the Schatten-von Neumann class S_p.