  The existence of cosmic ray particles up to the ultra-high energy limit (>
10^20 eV) is now beyond any doubt. The detection of cosmic particles with such
energies imposes a challenge for the comprehension of their sources and nature.
On one side, particles with such high energies are difficult to be produced by
any astrophysical source. On the other side, the interactions of these
particles with photons of the cosmic microwave background cause substantial
losses of energy which constraint the maximum distances that the particles are
able to travel from the sources to the detectors. Aiming to help to elucidate
the problem of UHECR source identification, we have performed 3-D simulations
of particle trajectories propagated through the stochastic intergalactic and an
extended Galactic halo magnetic fields. Going further, we have also performed
simulations of proton and Fe nuclei through the spiral Galactic magnetic field
(GMF) and built full-sky maps of their arrival direction distribution in both
the detector (after deflection in the GMF) and just outside the Galaxy. In this
work we summarize the main results of these investigations.
