The existence of astrophysical neutrinos with energies of tens of TeV and higher has been reliably established by the IceCube experiment; the first confirmations of this discovery are being obtained with the ANTARES and Baikal-GVD facilities. The observational results do not fully agree with what was expected before the start of these experiments. The origin of these neutrinos has not been conclusively established, and simple theoretical models, popular for decades, cannot explain all observational data. This review summarizes the experimental results with emphasis on those important for constraining theoretical models, discusses various scenarios for the origin of high-energy neutrinos and briefly lists particualr classes of their potential astrophysical sources. It is demonstrated that the observational data may be explained if the flux of astrophysical neutrinos includes the contribution of extragalactic sources, dominating at the highest energies, and the Galactic component, significant only at neutrino energies 100 TeV. Other possible scenarios are also discussed.CONTENTS * Invited review published in Physics Uspekhi in a special issue dedicated to the 50th anniversary of INR RAS, Russian