I will start by giving an introductory overview of different notions of generalized crystals (including quasicrystals), unified by the requirement that their Fourier transforms are atomic. Then we study the effect of random perturbations on the Fourier transform of such generalized crystal. The basic result I will present is that under mixing assumptions on the random perturbations, the Fourier transform of a random perturbation is almost surely equal to the Fourier transform of the unperturbed crystal, multiplied by the Fourier transform of the law of the noise. Thus for example for Gaussian i.i.d. perturbations with known law, the perturbed crystal's Fourier transform allows to recover the initial crystal. The case of (perturbations of) lattices is due to Yakir, and we weaken independence hypotheses and extend the theory to quasicrystals, and lattices in finite groups, Heisenberg groups, and other nilpotent groups. This is joint work with Rodolfo Viera from UC Chile.