IceCube is the first kilometer-scale neutrino detector. Built primarily for neutrino astronomy, it has recently discovered events with energies above 100 TeV that are likely to be from distant sources beyond the solar system. Among the events are three with deposited energies of more than 1 PeV, the highest-energy neutrinos ever detected. We review the astrophysical arguments that motivate such a large detector, and we describe how it works and how the high-energy events are reconstructed and identified above the background of atmospheric neutrinos. We also describe the broad range of neutrino physics and particle astrophysics topics addressed by IceCube, as well as its potential for the future.