High-energy (TeV-PeV) and ultra-high-energy (> 100 PeV) neutrinos of cosmic origin have a vast potential to probe neutrino physics in regimes of energy and distance otherwise unreachable. Notable examples include measuring the neutrino-matter cross section beyond accelerator energies, precision tests using the flavor composition of the neutrino flux, testing whether neutrinos are unstable, and whether they interact secretly with one another. Today, we are already tapping into this potential at high energies, thanks to recent discoveries by the IceCube neutrino telescope. In the coming decade, we will enter a regime of higher statistics and may extend our reach to ultra-high energies, thanks to an ambitious experimental program currently under planning. By means of illustration, I will briefly survey the rich landscape of high-energy physics with cosmic neutrinos, from the perspectives of theory and experiment. Along the way, I will point out manifest instances of the natural synergy between low-energy and high-energy neutrino experiments, and the need to account for astrophysical unknowns.