  We discuss the deep galaxy counts from the Hubble Deep Field (HDF) imaging
survey. At faint magnitudes, the slope of the differential number-magnitude
relation is flatter than 0.2 in all four HDF bandpasses. In the ultraviolet, a
fluctuation analysis shows that the flattening observed below U=26 mag is not
due to incompleteness and is more pronounced than in the other bands,
consistent with the idea that a redshift limit has been reached in the galaxy
distribution. A reddening trend of about 0.5 magnitude is observed at faint
fluxes in the colour-magnitude diagram, (U-V)_{eff} vs. V. We interpret these
results as the effect of intergalactic attenuation on distant galaxies. At flux
levels of AB=27 mag and in agreement with the fluctuation analysis and the
colour-magnitude relation, about 7% of the sources in U, 30% in B and 35% in V
are Lyman-break ``dropouts'', i.e. candidate star-forming galaxies at z>2. By
integrating the number counts to the limits of the HDF survey we find that the
mean surface brightness of the extragalactic sky is dominated by galaxies that
are relatively bright and are known to have <z>=0.6. To AB=29 mag, the
integrated light from resolved galaxies in the I-band is 2.1\pm 0.4 \times
10^{-20} erg/cm^2/s/Hz/sr, and its spectrum is well described by a broken
power-law. We discuss the predictions for the counts, colours, and luminosity
densities from standard low-q_0 pure-luminosity-evolution models without dust
obscuration, and find that they are unable to reproduce all the observed
properties of faint field galaxies.
