In this report we review the important progress made in recent years towards understanding the experimental data on ultra-high-energy ($E \gtrsim 10^9$ GeV) cosmic rays. We begin with a general survey of the available data, including a description of the energy spectrum, the nuclear composition, and the distribution of arrival directions. At this point we also give a synopsis of experimental techniques. After that, we introduce the fundamentals of cosmic ray acceleration and energy loss during propagation, with a view of discussing the conjectured nearby sources. Next, we survey the state of the art regarding the high- and ultra-high-energy cosmic neutrinos which may be produced in association with the observed cosmic rays. These neutrinos could constitute key messengers identifying currently unknown cosmic accelerators, possibly in the distant universe, because their propagation is not influenced by background photon or magnetic fields. Subsequently, we summarize the phenomenology of cosmic ray air showers. We describe the hadronic interaction models used to extrapolate results from collider data to ultra-high energies and the main electromagnetic processes that govern the longitudinal shower evolution. Armed with these two principal shower ingredients and motivation from the underlying physics, we describe the different methods proposed to distinguish the primary particle species. In the end, we explore how ultra-high-energy cosmic rays can be used as probes of beyond standard model physics models.

Matching version accepted for publication in Physics Reports. This article builds upon the content of arXiv:hep-ph/0206072, arXiv:astro-ph/0402371, arXiv:hep-ph/0407020, arXiv:1104.0509, arXiv:1307.5312, arXiv:1505.02153, arXiv:1709.07321, arXiv:1801.06160, and arXiv:1801.07170