  We present and compare in this paper new photometric redshift catalogs of the
galaxies in three public fields: the NTT Deep Field, the HDF-N and the HDF-S.
Photometric redshifts have been obtained for thewhole sample, by adopting a
$\chi^2$ minimization technique on a spectral library drawn from the Bruzual
and Charlot synthesis models, with the addition of dust and intergalactic
absorption. The accuracy, determined from 125 galaxies with known spectroscopic
redshifts, is $\sigma_z\sim 0.08 (0.3)$ in the redshift intervals $z=0-1.5
(1.5-3.5)$. The global redshift distribution of I-selected galaxies shows a
distinct peak at intermediate redshifts, z~0.6 at I_{AB}<26 and z~0.8 at
I_{AB}<27.5 followed by a tail extending to z~6. We also present for the first
time the redshift distribution of the total IR-selected sample to faint limits
($Ks \leq 21$ and $J\leq22$). It is found that the number density of galaxies
at 1.25<z<1.5 is ~ 0.1 /arcmin^22 at J<=21 and ~1./arcmin^2} at J<22, and drops
to 0.3/arcmin^2 (at J<22) at 1.5<z<2. The HDFs data sets are used to compare
the different results from color selection criteria and photometric redshifts
in detecting galaxies in the redshift range 3.5<z<4.5 Photometric redshifts
predict a number of high z candidates in both the HDF-N and HDF-S that is
nearly 2 times larger than color selection criteria, and it is shown that this
is primarily due to the inclusion of dusty models that were discarded in the
original color selection criteria by Madau et al 1998. In several cases, the
selection of these objects is made possible by the constraints from the IR
bands. Finally, it is shown that galactic M stars may mimic z>5 candidates in
the HDF filter set and that the 4 brightest candidates at $z>5$ in the HDF-S
are indeed most likely M stars. (ABRIDGED)
