We discuss the prospects for next generation neutrino telescopes, such as IceCube, to measure the flavor ratios of high-energy astrophysical neutrinos. The expected flavor ratios at the sources are φν e : φν µ : φν τ = 1 : 2 : 0, and neutrino oscillations quickly transform these to 1 : 1 : 1. The flavor ratios can be deduced from the relative rates of showers (νe charged-current, most ντ chargedcurrent, and all flavors neutral-current), muon tracks (νµ charged-current only), and tau lepton lollipops and double-bangs (ντ charged-current only). The peak sensitivities for these interactions are at different neutrino energies, but the flavor ratios can be reliably connected by a reasonable measurement of the spectrum shape. Measurement of the astrophysical neutrino flavor ratios tests the assumed production mechanism and also provides a very long baseline test of a number of exotic scenarios, including neutrino decay, CPT violation, and small-δm 2 oscillations to sterile neutrinos.PACS numbers: 95.85. Ry, 96.40.Tv, 13.35.Hb, 14.60.Pq