  Precision measurements of the cosmic microwave background temperature
anisotropy on scales $\ell > 500$ will be available in the near future.
Successful interpretation of these data is dependent on a detailed
understanding of the damping tail and cosmological recombination of both
hydrogen and helium. This paper and two companion papers are devoted to a
precise calculation of helium recombination. We discuss several aspects of the
standard recombination picture, and then include feedback, radiative transfer
in HeI lines with partial redistribution, and continuum opacity from HI
photoionization. In agreement with past calculations, we find that HeII
recombination proceeds in Saha equilibrium, whereas HeI recombination is
delayed relative to Saha due to the low rates connecting excited states of HeI
to the ground state. However, we find that at $z<2200$ the continuum absorption
by the rapidly increasing HI population becomes effective at destroying photons
in the HeI $2^1P^o-1^1S$ line, causing HeI recombination to finish around
$z\approx 1800$, much earlier than previously estimated.

Astrophysics