Ultra-high-energy cosmic rays (UHECRs) hit the Earth's atmosphere with energies exceeding 10(18)eV. This is the same energy as carried by a tennis ball moving at 100 km h-1, but concentrated on a subatomic particle. UHECRs are so rare (the flux of particles with E>10(20)eV is 0.5 km -2 per century) that only a few such particles have been detected over the past 50 years. Recently, the HiRes and Auger experiments have reported the discovery of a high-energy cut-off in the UHECR spectrum, and Auger has found an apparent clustering of the highest energy events towards nearby active galactic nuclei. Consensus is building that the highest energy particles are accelerated within the radio-bright lobes of these objects, but it remains unclear how this actually happens, and whether the cut-off is due to propagation effects or reflects an intrinsically physical limitation of the acceleration process. The low event statistics presently allows for many different plausible models; nevertheless observations are beginning to impose strong constraints on them. These observations have also motivated suggestions that new physics may be implicated. We present a review of the key theoretical and observational issues related to the processes of propagation and acceleration of UHECRs and proposed solutions.