  The astrophysical neutrinos discovered by IceCube have the highest detected
neutrino energies --- from TeV to PeV --- and likely 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 astrophysical unknowns. We 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, larger
neutrino energies --- up to tens of EeV --- could be observed with larger
detectors and further our reach.
