The astrophysical neutrinos recently discovered by the IceCube neutrino telescope have the highest detected neutrino energies --- from TeV to PeV --- and travel the longest distances --- up to a few Gpc, the size of the observable Universe. These features make them naturally attractive probes of fundamental particle-physics properties, possibly tiny in size, at energy scales unreachable by any other means. The decades before the IceCube discovery saw many proposals of particle-physics studies in this direction. Today, those proposals have become a reality, in spite of prevalent astrophysical unknowns. We showcase examples of studying fundamental neutrino physics at these scales, including some of the most stringent tests of physics beyond the Standard Model.