The astrophysical neutrinos recently discovered by IceCube 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.  I will showcase examples of doing fundamental neutrino physics at these scales, including some of the most stringent tests of physics beyond the Standard Model.  In the future, higher neutrino energies --- up to tens of EeV --- larger detectors, and improved detection techniques will further our reach.