  We have studied the X-ray source population of the face-on spiral galaxy M101
(NGC 5457). Within a field of radius 17' (36 kpc at the distance of 7.2 Mpc),
covered by an ultradeep (229 ks) ROSAT HRI image, 51 X-ray sources are detected
with signal-to-noise ratios greater than 3.5. About half of these sources are
associated with the galaxy. We determine the average luminosity distribution of
the sources.
  We have examined spatial, spectral, and timing properties of the X-ray
sources. In particular, we have explored the nature of various superluminous
X-ray sources with luminosities significantly greater than the Eddington limit
for a $\sim 1.6 M_\odot$ object (neutron star). These X-ray sources, detected
in various ROSAT HRI and PSPC observations, are not transients and appear to
result from recent massive star formation in outer spiral arms. 3 superluminous
PSPC sources are associated with giant HII complexes and are clearly resolved.
2 other superluminous RHRI sources are likely associated with shell-like
supernova (or more likely hypernova) remnants, which are known to be abnormally
luminous in optical and/or radio. We further identify 2 superluminous sources,
which all show highly-absorbed
  X-ray spectra and time variability during and/or between the observations, as
candidates for X-ray binary systems that contain black holes.
  A comparison of 7 nearby spirals shows that their X-ray source luminosity
distributions, normalized by total HI masses, are very similar. But both the
number of superluminous X-ray sources and the total X-ray luminosity appear to
be correlated with the star forming rate of a galaxy.
