The discovery using ASCA that the featureless X-ray spectrum from the shell
of SN 1006 is produced by shock-accelerated TeV electrons, and is thus an
indicator of cosmic ray acceleration, has sparked a search for X-ray spectr
al evidence for cosmic ray acceleration in other remnants, as well as much
theoretical work. Thus far evidence for cosmic ray acceleration, in the for
m of synchrotron continuum components, has been found in several additional
remnants, including Cas A, G347.5-0.5, and IC 443. The information provide
d by both detections and non-detections of these components can be used to
investigate the shock acceleration process. We review the latest results of
the continuing search for X-ray indicators of cosmic ray acceleration in G
alactic and Magellanic Cloud SNR's and what we have learned thus far from i
t.