  The study of ultra-high energy cosmic rays (UHECR) at Earth cannot prescind
from the study of their propagation in the Universe. In this paper, we present
HERMES, the \emph{ad hoc} Monte Carlo code we have developed for the realistic
simulation of UHECR propagation. We discuss the modeling adopted to simulate
the cosmology, the magnetic fields, the interactions with relic photons and the
production of secondary particles. In order to show the potential applications
of HERMES for astroparticle studies, we provide an estimation of the surviving
probability of UHE protons, the GZK horizons of nuclei and the all-particle
spectrum observed at Earth in different astrophysical scenarios. Finally, we
show the expected arrival direction distribution of UHECR produced from nearby
candidate sources. A stable version of HERMES will be released in the next
future for public use together with libraries of already propagated nuclei to
allow the community to perform mass composition and energy spectrum analysis
with our simulator.
