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A hyperuniform many-body system is characterized by a structure factor \(S(k)\) that vanishes in the small-wavenumber limit or equivalently by a local number variance \(\sigma_N^2(R)\) associated with a spherical window of radius \(R\) that grows more slowly than Rd in the large-R limit. Thus, the hyperuniformity implies anomalous suppression of long-wavelength density fluctuations relative to those in typical disordered systems, i.e., \(\sigma_N ^2(R)\sim R^d\) as \(R\to\infty\). Hyperuniform systems include perfect crystals, quasicrystals, and special disordered systems. Disordered hyperuniform systems are amorphous states of matter that lie between a liquid and crystal [S. Torquato *et al*., Phys. Rev. X 5, 021020 (2015)], and have been the subject of many recent investigations due to their novel properties. In the same way that there is no perfect crystal in practice due to the inevitable presence of imperfections, such as vacancies and dislocations, there is no “perfect” hyperuniform system, whether it is ordered or not. Thus, it is practically and theoretically important to quantitatively understand the extent to which imperfections introduced in a perfectly hyperuniform system can degrade or destroy its hyperuniformity and corresponding physical properties. This paper begins such a program by deriving explicit formulas for \(S(k)\) in the small-wavenumber regime for three types of imperfections: (1) uncorrelated point defects, including vacancies and interstitials, (2) stochastic particle displacements, and (3) thermal excitations in the classical harmonic regime. We demonstrate that our results are in excellent agreement with numerical simulations. We find that “uncorrelated” vacancies or interstitials destroy hyperuniformity in proportion to the defect concentration p. We show that “uncorrelated” stochastic displacements in perfect lattices can never destroy the hyperuniformity but it can be degraded such that the perturbed lattices fall into class III hyperuniform systems, where \(\sigma_N ^2(R)\sim R^{d-\alpha}\) as \(R\to\infty\) and \(0<\alpha<1\). By contrast, we demonstrate that certain “correlated” displacements can make systems nonhyperuniform. For a perfect (ground-state) crystal at zero temperature, increase in temperature \(T\) introduces such correlated displacements resulting from thermal excitations, and thus the thermalized crystal becomes nonhyperuniform, even at an arbitrarily low temperature. Our work provides the theoretical underpinnings to systematically study the effect of imperfections on the physical properties of hyperuniform materials.