  Chandra observations of nearby galaxies have revealed a number of X-ray
sources characterized by high luminosities ($L_X > \sim10^{36}$ erg s$^{-1}$)
and $k T$ in the range $100 - 350$ eV. These ``quasisoft X-ray sources'' (QSSs)
are harder than luminous supersoft X-ray sources (SSSs), whose characteristic
temperatures are tens of eV, but, with little or no emission above 2 keV, they
are significantly softer than most canonical X-ray sources. They are likely to
include a range of physical systems; some may be common systems in unusual
states: neutron stars or stellar-mass black holes (BHs), or SNRs. Others may be
accreting BHs of intermediate mass. We have analyzed {\it Chandra} data from
$19 $ near-by galaxies to identify QSSs with sufficiently high count rates to
allow spectral fits. Six of these galaxies have been studied in great detail;
in these we find 89 SSSs and 122 QSSs. In this paper we present spectra for
QSSs those with more than 50 counts. We also use data from six of these
galaxies to study the broadband spectral distributions of QSSs and to compare
them with those of other X-ray sources. Since QSSs in the Milky Way are likely
to be missed by our selection procedures, we discuss other signatures by which
they can be found. If some of the QSSs for which we have spectra are accreting
BHs, the lower bounds for the BH mass range from roughly $30 M_\odot$ to $1000
M_\odot$.
