  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.
