  We present a systematic analysis of archival HST WFPC2 ``Association'' data
sets that correlate with the Chandra positions of a set of 44 ultra-luminous
X-ray sources (ULXs) of nearby galaxies. We have improved the Chandra-HST
relative astrometry whenever possible. Disparate numbers of potential ULX
counterparts are found, and in some cases none are found. The lack of or low
number of counterparts in some cases may be due to insufficient depth in the
WFPC2 images. Particularly in late-type galaxies, the HST image in the ULX
region was often complex or crowded. We therefore address various scenarios for
the nature of the ULX since it is not known which, if any, of the sources found
are true counterparts. The optical luminosities of the sources are typically in
the range 10^4-6 L_sun. In several cases color information is available, with
the colors roughly tending to be more red in early-type galaxies. This suggests
that, in general, the (potential) counterparts found in early-type galaxies are
likely to be older stellar populations, and are probably globular clusters.
Several early-type galaxy counterparts have blue colors, which may be due to
younger stellar populations in the host galaxies, however these could also be
background sources. In spiral galaxies the sources may also be due to localized
structure in the disks rather than bound stellar systems. Alternatively some of
the counterparts in late-type galaxies may be isolated supergiant stars. The
observed X-ray/optical flux ratio is diluted by the optical emission of the
cluster in cases where the system is an X-ray binary in a cluster, particularly
in the case of a low-mass X-ray binaries in old cluster. (abridged)
