  A number of nearby Northern Hemisphere shell-type Supernova Remnants (SNRs)
has been observed in TeV gamma rays, but none of them could be detected so far.
This failure calls for a critical reevaluation of the theoretical arguments for
gamma-ray emission of SNRs. The present paper discusses diffusive shock
acceleration in shell-type SNRs in full kinetic theory. Emphasis is also given
to the possible problems for VHE gamma-ray production due to the environmental
conditions a SN progenitor finds itself in. Observational upper limits are
compared with theoretical predictions for the gamma-ray flux. Empirical
arguments from the observation of X-ray power law continua for electron-induced
Inverse Compton gamma-ray emission at TeV energies are discussed in their
relation to the nucleonic Pi-zero decay emission from the same objects.
Finally, a point is made for the simplest case of SNe Ia, expected to explode
in a uniform circumstellar medium. Here in particular the very recently
detected Southern Hemisphere remnant of SN 1006 is compared with Tycho's SNR.
On the basis of the assumed parameters for the two remnants SN 1006 is
tentatively identified with a remnant whose TeV gamma-ray emission is dominated
by Inverse Compton radiation. Tycho might be dominantly a Pi-zero decay
gamma-ray source.
