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We give a survey of recent advances in the theory of spaces of entire functions related to the notion of spectral synthesis. In particular, we discuss a solution of a longstanding problem about spectral synthesis for systems of exponentials in as well as its generalization to de Branges spaces of entire functions. In the de Branges space setting the problem can be related (via a functional model) to spectral theory of rank one perturbations of compact selfadjoint operators; this leads to unexpected examples of rank one perturbations which do not admit spectral synthesis. Related problems for Fock-type spaces are also considered.