  Data from {\it Chandra} observations of thirty nearby galaxies were analyzed
and 365 X-ray point sources were chosen whose spectra were not contaminated by
excessive diffuse emission and not affected by photon pile up. The spectra of
these sources were fitted using two spectral models (an absorbed power-law and
a disk blackbody) to ascertain the dependence of estimated parameters on the
spectral model used. It was found that the cumulative luminosity function
depends on the choice of the spectral model, especially for luminosities $>
10^{40}$ ergs/s. In accordance with previous results, a large number ($\sim
80$) of the sources have luminosities $> 10^{39}$ ergs/s (Ultra-Luminous X-ray
sources) with indistinguishable average spectral parameters (inner disk
temperature $\sim 1$ keV and/or photon index $\Gamma \sim 2$) with those of the
lower luminosities ones. After considering foreground stars and known
background AGN,we identify four sources whose minimum luminosity exceed
$10^{40}$ ergs/s, and call them Extremely Luminous X-ray sources (ELX). The
spectra of these sources are in general better represented by the disk black
body model than the power-law one. These ELX can be grouped into two distinct
spectral classes. Two of them have an inner disk temperature of $< 0.5$ keV and
hence are called ``supersoft'' ELX, while the other two have temperatures
$\gtrsim 1.3$ keV and are called ``hard'' ELX. The estimated inner disk
temperatures of the supersoft ELX are compatible with the hypothesis that they
harbor intermediate size black holes, which are accreting at $\sim 0.5$ times
their Eddington Luminosity. The radiative mechanism for hard ELX, seems to be
Inverse Comptonization, which in contrast to standard black holes systems, is
probably saturated.
