The observation of twenty cosmic-ray air-showers at and above1020&hthinsp;eVposes immediate problems for particle astrophysics: how the primary particles are accelerated to these energies, and how the primaries get here through the 2.7 K microwave background filling the Universe. In addition, the highest-energy events exhibit near-isotropy in the large, but a surprising small-scale clustering on the celestial sky. An overview of the puzzles is presented, followed by a discussion of many of the models proposed to solve these puzzles. Emphasis is placed on (i) the role neutrino primaries may play in resolving issues, and (ii) the natural manner in which hypothetical magnetic monopoles solve the acceleration and propagation problems. Finally, the many signatures by which cosmic ray experiments in the near future will discriminate among the proposed models are presented. ©2001 American Institute of Physics.