A short introduction to the theory of neutrino electromagnetic properties is presented. Then the main issues related to studies of neutrino electromagnetic characteristics in the scattering experiments are considered. A set of basic neutrino electromagnetic processes that can occur in various astrophysical conditions and be fundamentally observable are discussed. The future prospects that open up in the study of the electromagnetic properties of neutrinos are also discussed, including those provided by the recently obtained results of the XENON1T experiment, new $GEMMA-\nu Gen$ experiment and the work begun on the preparation of a new experiment to detect coherent elastic $\nu$-helium atom scattering with use of a high-intensity tritium antineutrino source in Sarov.