  Absorption due to HeII Ly-alpha has been detected at low resolution in the
spectra of four quasars between redshifts z = 2.74-3.29. We assess these
observations, giving particular attention to the radiative transfer of the
ionizing background radiation, cloud diffuse emission and ionization physics,
and statistical fluctuations. We use high-resolution observations of HI
absorption towards quasars to derive improved models for the opacity of the
intergalactic medium (IGM) and use these models to calculate the HI and HeII
photoionization history in both optically-thin and self-shielded clouds, and
the average line-blanketing contribution to HeII absorption. The derived
ionization rate at z = 2-4, Gamma(HI) = (1-3)x10^-12 s^-1, is consistent with
the ionizing background intensity inferred from the proximity effect, but it
remains larger than that inferred by N-body hydrodynamical simulations of the
Lya absorber distribution. The HeII observations are con- sistent with line
blanketing from clouds with N(HI) > 10^12 cm^-2, although a contribution from a
more diffuse IGM would help to explain the observed opacity. We compute the
expected HeII optical depth and examine the sizable fluc- tuations that arise
from variations in the cloud numbers and ionizing radiation field. We assess
how HeII absorption constrains the intensity and spectrum of the ionizing
radiation and the fractional contributions of the dominant sources (quasars and
starburst galaxies). Finally, we demonstrate how high-resolution UV
observations can distinguish between absorption from the diffuse IGM and the
Lya forest clouds and determine the source of the ionizing background.
