  Quillen's fundamental spectral sequences relate Andr\'{e}-Quillen homology
and cohomology to Tor and Ext functors. The five-term exact sequences arising
from these spectral sequences are leveraged to characterize regular and
complete intersection local rings. Despite their immense importance in the
theory of Andr\'{e}-Quillen homology and cohomology, these spectral sequences
are not treated in the literature as they merit. A sketchy argument with gaps
for a special case of the first spectral sequence appears in an unpublished
manuscript of Quillen, while the second spectral sequence is only stated
without proof in another paper of Quillen. A rigorous proof of the spectral
sequences requires a delicate investigation of the subtle structures involved.
The goal of this expository article is to present a comprehensive and detailed
proof of Quillen's fundamental spectral sequences in a readable and
well-structured manner.
