  We describe our spectroscopic follow-up to the Local Group Galaxy Survey
(LGGS) photometry of M31 and M33. We have obtained new spectroscopy of 1895
stars, allowing us to classify 1496 of them for the first time. Our study has
identified many foreground stars, and established membership for hundreds of
early- and mid-type supergiants. We have also found 9 new candidate Luminous
Blue Variables and a previously unrecognized Wolf-Rayet star. We republish the
LGGS M31 and M33 catalogs with improved coordinates and including spectroscopy
from the literature and our new results. The spectroscopy in this paper is
responsible for the vast majority of the stellar classifications in these two
nearby spiral neighbors. The most luminous (and hence massive) of the stars in
our sample are early-type B supergiants, as expected; the more massive O stars
will be fainter visually, and thus mostly remain unobserved so far. The
majority of the unevolved stars in our sample are in the 20-40Mo range.
