In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] 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 paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R2), 0<α<1, of functions of two variables, and N1 and N2 are normal operators, then {double pipe}f(N1){double pipe}f(N2){double pipe}≤const{double pipe}f{double pipe}Λα{double pipe}N1-N2{double pipe}α. We obtain a more general result for functions in the space Λω(R2)={f:|f(ζ1)-f(ζ2)|≤constω(|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., {double pipe}f(N1)-f(N2){double pipe}≤const{double pipe}f{double pipe}B∞11{double pipe}N1-N2{double pipe}. We also study properties of f(N1)-f(N2) in the case when f∈Λα(R2) and N1-N2 belongs to the Schatten-von Neumann class Sp.