To investigate the properties of Ultra Luminous compact X-ray sources (ULXs), nine spiral galaxies, containing twelve ULXs, were observed with ASCA. For comparison, three luminous extragalactic supernova remnants (SNRs) were also studied. Most of the objects are consistent with point-sources within the spatial resolution of ROSAT HRI. ASCA spectra of SNRs and ULXs show a clear difference; the former show the features of an optically-thin hot plasma, whereas the latter lack such spectral properties, implying that a ULX is not an unidentified SNR. Among twelve ULXs, nine exhibit spectra which can be represented with a so-called multi-color disk blackbody (MCD) emission describing optically-thick multi-temperature emission from a standard accretion disk; one shows spectrum which can be expressed with the MCD plus a power-law model; and two have power-law spectra of photon index ~ 1.5. These spectral properties suggest that ULXs are luminous extragalactic mass-accreting black-hole binaries. The bolometric luminosities of the MCD emission are 1039 - 1040 erg s-1, requiring high black-hole mass up to ~ 100 Msolar to sustain sub-Eddington radiation. However, the observed inner-most disk temperatures, ranging 1.0-1.8 keV, are too high to account for the inferred high black-hole mass if we assume a standard accretion disk around a Schwarzschild black hole. Changes of the inner-disk radius have also been detected from three ULXs, which contradict the properties of Galactic and Magellanic black-hole binaries. Several attempts were made to solve these problems. Finally the interpretation of ULXs as emission from an optically-thick accretion disk around a rapidly rotating black hole is proposed.