  We discuss the intergalactic propagation of ultra-high-energy cosmic rays
(UHECRs) with energies E \geq 10^{18.5} eV. We consider the propagation of
UHECRs under the influence of the energy-dependent deflection by a weak random
magnetic field in the intergalactic medium and energy losses by photo-pion and
pair production. We calculate arrival spectra taking full account of the
kinematics of photo-pion production and the Poisson statistics of the
photo-pion interaction rate.
  We give estimates for the deflection of UHECRs from the line of sight to the
source, time delays with respect to photons from the same source, arrival
spectra and source statistics. These estimates are confirmed by numerical
simulations of the propagation in energy evolution of UHECRs. These simulations
demonstrate that the often-used continuous approximation in the treatment of
energy losses due to photo-pion production on the cosmic microwave background
(CMWB) cannot be justified for UHECRs.
  We discuss the implications of these results for the observed flux of
particles above the Greisen-Zatsepin-Kuz'min cut-off in two of the scenarios
that have been proposed for the production of these particles: continuous
production in the large shock waves associated with powerful radio galaxies, or
possibly large-scale structure formation, and the impulsive production at
relativistic blast waves associated with cosmological gamma-ray bursts.
