  We present the results of several collisional-radiative models describing
optically-thin emissivities of the main lines in neutral helium formed by
recombination, for a grid of electron temperatures and densities, typical of H
II regions and Planetary Nebulae. Accurate emissivities are required for
example to measure the helium abundance in nebulae and as a consequence its
primordial value. We compare our results with those obtained by previous
models, finding significant differences, well above the target accuracy of one
percent. We discuss in some detail our chosen set of atomic rates and the
differences with those adopted by previous models. The main differences lie in
the treatment of electron and proton collision rates and we discuss which
transitions are least sensitive to the choice of these rates and therefore best
suited to high precision abundance determinations. We have focused our
comparisons on the case B approximation where only He and He$^+$ are
considered, but also present results of full models including the bare nuclei,
photo-excitation and photo-ionisation and either black-body or observed
illuminating spectrum in the case of the Orion nebula, to indicate which
spectral lines are affected by opacity. For those transitions, accurate
radiative transfer calculations should be performed. We provide tables of
emissivities for all transitions within $n \le 5$ and all those between the $n
\le 5$ and $n' \le 25$ states, in the log $T_{\rm e}$ [K]=10$^{3.0(0.1)4.6}$
and log $N_{\rm e}$ [cm$^{-3}$]=10$^{2(0.5)6}$ ranges, and a FORTRAN code to
interpolate to any $T_{\rm e}, N_{\rm e}$ within these ranges.
