  We have developed an implicit, multi-group, time-dependent, spherical
neutrino transport code based on the Feautrier variables, the tangent-ray
method, and accelerated ${\bf \Lambda}$ iteration. The code achieves high
angular resolution, is good to O($v/c$), is equivalent to a Boltzmann solver
(without gravitational redshifts), and solves the transport equation at all
optical depths with precision. In this paper, we present our formulation of the
relevant numerics and microphysics and explore protoneutron star atmospheres
for snapshot post-bounce models. Our major focus is on spectra, neutrino-matter
heating rates, Eddington factors, angular distributions, and phase-space
occupancies. In addition, we investigate the influence on neutrino spectra and
heating of final-state electron blocking, stimulated absorption, velocity terms
in the transport equation, neutrino-nucleon scattering asymmetry, and weak
magnetism and recoil effects. Furthermore, we compare the emergent spectra and
heating rates obtained using full transport with those obtained using
representative flux-limited transport formulations to gauge their accuracy and
viability. Finally, we derive useful formulae for the neutrino source strength
due to nucleon-nucleon bremsstrahlung and determine bremsstrahlung's influence
on the emergent $\nu_{\mu}$ and $\nu_{\tau}$ neutrino spectra.
