IceCube has observed astrophysical neutrinos with energies up to 10 PeV (1 PeV = 10^15 eV)), 20,000 times higher in energy than are studied with particle accelerators.  They allow us to extend studies of neutrino interactions by many orders of magnitude upward in energy.  I will present two specific measurements: a measurement of the neutrino interaction cross-section, determined by measuring neutrino absorption in the Earth, and a measurement of the inelasticity, the fraction of neutrino energy transferred to the outgoing lepton.   In addition to probing neutrino physics and searching for new (beyond-standard-model) physics, these studies also allow us to make better measurements of the high-energy astrophysical neutrino flux.    I will also discuss planned future measurements, using a new technique: looking for coherent radio Cherenkov emission from ultra-high energy neutrino showers.  A new radio-array would instrument 100 km^3 of Antarctic ice, and could extend these studies up to 10^20 eV – beyond the energy reach of CERN’s Large Hadron Collider