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The question of the origin of ultra-high-energy cosmic rays at relativisticshock waves is discussed in the light of results of recent Monte Carlo studiesof the first-order Fermi particle acceleration (Niemiec & Ostrowski 2006,Niemiec et al. 2006). The models of the turbulent magnetic field near the shockconsidered in these simulations include realistic features of the perturbedmagnetic field structures at the shock, which allow us to study all the fieldand particle motion characteristics that are important for cosmic-rayacceleration. Our results show that turbulent conditions near the shock, thatare consistent with the shock jump conditions, lead to substantialmodifications of the acceleration process with respect to the simplifiedmodels, that produce wide-range power-law energy distributions, often with the"universal" spectral index. Relativistic shocks are essentially alwayssuperluminal, and thus they preferentially generate steep particle spectra withcutoffs well below the maximum scattering energy, often not exceeding theenergy of the compressed background plasma ions. Thus, cosmic-ray accelerationto very high energies at relativistic shock waves is inefficient, and suchshocks are not expected to be the sources of ultra-high-energy particles.

The inefficiency of the first-order Fermi process in UHECR production at relativistic shocks