It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λ_α(R{double-struck}) with 0<α<1, then {norm of matrix}f(A)-f(B){norm of matrix}≤const{norm of matrix}A-B{norm of matrix}^α for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ₁(R{double-struck}): for arbitrary self-adjoint operators A and K we have {norm of matrix}f(A-K)-2f(A)+f(A+K){norm of matrix}≤const{norm of matrix}K{norm of matrix}. We also obtain analogs of this result for all Hölder-Zygmund classes Λ_α(R{double-struck}), α>0. Then we find a sharp estimate for {norm of matrix}f(A)-f(B){norm of matrix} for functions f of class Λ_ω=def{f:ω_f(δ)≤constω(δ)} for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which {norm of matrix}f(A)-f(B){norm of matrix}≤constω({norm of matrix}A-B{norm of matrix}) for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q-Qf(A) and quasicommutators f(A)Q-Qf(B). Finally, we estimate the norms of finite differences ∑_j=0^m(-1)^m-j(^m_j)f(A+jK) for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.