  We combine data from a number of N-body simulations to predict the abundance
of dark halos in Cold Dark Matter universes over more than 4 orders of
magnitude in mass. A comparison of different simulations suggests that the
dominant uncertainty in our results is systematic and is smaller than 10--30%
at all masses, depending on the halo definition used. In particular, our
``Hubble Volume'' simulations of \tcdm and \lcdm cosmologies allow the
abundance of massive clusters to be predicted with uncertainties well below
those expected in all currently planned observational surveys. We show that for
a range of CDM cosmologies and for a suitable halo definition, the simulated
mass function is almost independent of epoch, of cosmological parameters, and
of initial power spectrum when expressed in appropriate variables. This
universality is of exactly the kind predicted by the familiar Press-Schechter
model, although this model predicts a mass function shape which differs from
our numerical results, overestimating the abundance of ``typical'' halos and
underestimating that of massive systems.
