How, where and when do cosmic rays reach ultrahigh energies?

Abstract: Understanding the origins of ultrahigh energy cosmic rays (UHECRs) - which reach energies in excess of $10^{20}~{\rm eV}$ - stretches particle acceleration physics to its very limits. In this review, we discuss how such energies can be reached, using general arguments that can often be derived on the back of an envelope. We explore possible particle acceleration mechanisms, with special attention paid to shock acceleration. Informed by the arguments derived, we discuss where UHECRs might come from and which classes of powerful astrophysical objects could be UHECR sources; generally, we favour radio galaxies, GRB afterglows and other sources which are not too compact and dissipate prodigious amounts of energy on large scales, allowing them to generate large products $\beta B R$ without the CRs undergoing restrictive losses. Finally, we discuss when UHECRs are accelerated by highlighting the importance of source variability, and explore the intriguing possibility that the UHECR arrival directions are partly a result of "echoes" from magnetic structures in the local Universe.