Abstract: In perturbation theory, the cyclicity (i.e. completeness of the forward orbit for some vector) of an operator is often a natural assumption
in the hypothesis of many results. We discuss two sets of results. (a) We consider rank one perturbations and some applications concerning the
locations where functions of the Payley-Wiener class are allowed to have zeros. (b) We study so-called Anderson-type Hamiltonians (a
generalization of random Schroedinger operators) and state that under mild conditions, every non-zero vector is cyclic almost surely. This result is
related to the long-standing problem of Anderson localization.