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A review of the theory and phenomenology of neutrino electromagnetic properties is presented. It is shown that studies of neutrino electromagnetic properties provide powerful tools to probe the physics beyond the Standard Model. A massive neutrino even in the easiest generalization of the Standard Model inevitably has nonzero electromagnetic characteristics, at least a nonzero magnetic moment. Although its value is determined by the neutrino mass and is, in general, very small, in other BSM theories, as for example the supersymmetric models, much larger values of magnetic moments are predicted. A discussion on a derivation of the general structure of the electromagnetic interactions of Dirac and Majorana neutrinos is given. Then we discuss experimental constraints on neutrino magnetic and electric dipole moments, electric millicharge, charge radius and anapole moments from the terrestrial laboratory experiments. The best world experimental bounds on neutrino electromagnetic properties are confronted with the predictions of theories beyond the Standard Model. A special credit is done to bounds on neutrino magnetic moments obtained by the reactor MUNU, GEMMA and TEXONO and solar (Super-Kamiokande and Borexino) experiments. The effects of neutrino electromagnetic interactions in astrophysical environments are also reviewed. The main manifestation of neutrino electromagnetic interactions, such as: 1) the radiative decay in vacuum, in matter and in a magnetic field, 2) the Cherenkov radiation, 3) the plasmon decay, 4) spin light in matter, 5) spin and spin-flavour precession, 6) neutrino pair production in a strong magnetic field, and the related processes along with their astrophysical phenomenology are also considered. On this basis, the astrophysical constraints on neutrino electromagnetic characteristics are reviewed. Possibilities in improving the terrestrial laboratory bounds on neutrino electromagnetic properties are considered. In this concern, new reactor neutrino scattering experiments, that are expected to be realized in the near future, are discussed. The related problem of the coherent elastic neutrino-nucleus scattering, which has not been experimentally observed so far, but which is expected to be accessible in the reactor experiments when lowering the energy threshold of the employed detectors, is also touched upon. Finally, it is shown that observations of solar and supernova neutrinos in the future water-Cherenkov detector Hyper-Kamiokande and liquid-scintillator detectors SNO+, JUNO, RENO50 and LENA will greatly improve current knowledge about the electromagnetic properties of neutrinos. References 1. C. Guinti and A. Studenikin, “Neutrino electromagnetic interactions: a window to new physics”, Rev. Mod. Phys. 87 (2015) 531-591. 2. A. Studenikin, “New bounds on neutrino electric millicharge from limits on neutrino magnetic moment”, Europhys. Lett. 107 (2014) 21001. 3. A. Studenikin, I. Tokarev, “Millicharged neutrino with anomalous magnetic moment in rotating magnetized matter”, Nucl. Phys. B 884 (2014) 396-407. 4. K. Kouzakov, A. Studenikin, “Theory of neutrino-atom collisions: the history, present status and BSM physics”, Adv. High Energy Phys. 2014 (2014) 569409 (16 p.). 5. I. Balantsev, A. Studenikin, “From electromagnetic neutrinos to new electromagnetic radiation mechanism in neutrino fluxes”, Int. J. Mod. Phys. A30 (2015) 1530044 (10 p) . 6. C. Giunti, K. Kouzakov, Y.-F. Li, A. Lokhov, A. Studenikin, Sh. Zhou, “Electromagnetic neutrinos in laboratory experiments and astrophysics”, Ann. Phys. 528 (2016) 198-215. 7. K. Kouzakov, A. Studenikin, “Neutrino-atom collisions”, arXiv: 1603.02462.