2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226731This is a survey article. We consider different problems in connection with the behavior of functions of operators under perturbations of operators. We deal with three classes of operators: unitary operators, self-adjoint operators, and contractions. We study operator Lipschitz and operator differentiable functions. We also study the behavior of functions under perturbations of an operator by an operator of Schatten--von Neumann class $\bS_p$ and apply the results to the Livschits--Krein and Koplienko--Neidhardt trace formulae. We also include in this survey article recent unexpected results obtained in a joint paper with Aleksandrov on operator Hölder--Zygmund functions.37 pagesFunctional AnalysisClassical Analysis and ODEsComplex VariablesSpectral Theory47A55, 47B49, 47A60text