Shell-like supernova remnants (SNRs) are primary candidates for the origin of Galactic cosmic rays. However, among the known SNRs (about 220), only a small fraction has been known to exhibit the synchrotron X-ray spectrum, that is considered to be a piece of evidence for high energy particle acceleration. Synchrotron X-ray emitting SNRs are known to be systematically radio-quiet compared to the SNRs that do not emit synchrotron X-rays. Therefore, most synchrotron X-ray emitting SNR candidates may have escaped detection in the previous systematic radio surveys. On the other hand, hard X-ray surveys are effective to search for synchrotron X-ray emitting SNRs, because of its penetration power. Thus we have searched for SNRs in the ASCA Galactic Plane Survey data, the first Galactic imaging survey in $>$2 keV, and discovered 14 candidates. Deep follow-up observations with ASCA, XMM, or Chandra on 5 of them revealed 2 sources to be synchrotron X-ray emitting SNRs. Furthermore we confirmed non-thermal X-ray spectra from the other 3 sources, though the origin is yet unknown. We report the observational results and discuss the X-ray origin.