We study sub-relativistic and ultra-relativistic components of high-energy particles associated with supernova remnants (SNRs) using X-ray imaging and spectroscopic observations with ASCA and Chandra. Based on the extensive analysis of two shell-type SNRs, gamma Cygni and RX J1713.7−3946, we have found new X-ray features in SNRs, namely (1) very flat-spectrum X-ray emission, and (2) unexpectedly complex structures in synchrotron X-ray images. We propose that the very flat X-ray spectrum is the characteristic bremsstrahlung X-ray spectrum, providing us a new diagnostic tool to study the largely-unknown component of low-energy cosmic rays in the Galaxy. Also we argue that the resolved X-ray features in the synchrotron X-ray images may challenge the perceptions of standard (diffusive shock-acceleration) models concerning the production, propagation and radiation of relativistic particles in supernova remnants.