High-energy astrophysical neutrinos have been observed by several telescopes in the last decade, but their sources still remained unknown. We address the problem of locating astrophysical neutrinos’ sources in a statistical manner. We show that blazars positionally associated with IceCube neutrino detections have stronger parsec-scale radio cores than the rest of the sample. The probability of a chance coincidence is only 4*10^-5 corresponding to a significance of 4.1 sigma. We explicitly list four strong radio blazars as highly probable sources of neutrinos above 200 TeV: 3C 279, NRAO 530, PKS 1741-$038, and PKS 2145+067. There are at least 70 more radio-bright blazars that emit neutrinos of lower energies starting from TeVs. Using continuous RATAN-600 monitoring of VLBI-selected blazars, we find that radio flares at frequencies above 10 GHz coincide with neutrino arrival dates. The most pronounced example of such behavior is PKS 1502+106 that experienced a major flare in 2019. We demonstrate that the majority of IceCube astrophysical neutrino flux derived from muon-track analyses may be explained by blazars, that is AGNs with bright Doppler-boosted jets. High-energy neutrinos can be produced in photohadronic interactions within parsec-scale relativistic jets. Radio-bright blazars associated with neutrino detections have very diverse gamma-ray properties, which suggests that gamma-rays and neutrinos may be produced in different regions of blazars and not directly related. A narrow jet viewing angle is, however, required to detect either of them. We conclude with discussion of recent independent tests and extensions of our findings.