  We deal with kernel theorems for modulation spaces. We completely
characterize the continuity of a linear operator on the modulation spaces $M^p$
for every $1\leq p\leq\infty$, by the membership of its kernel to (mixed)
modulation spaces. Whereas Feichtinger's kernel theorem (which we recapture as
a special case) is the modulation space counterpart of Schwartz' kernel theorem
for temperate distributions, our results do not have a couterpart in
distribution theory. This reveals the superiority, in some respects, of the
modulation space formalism upon distribution theory, as already emphasized in
Feichtinger's manifesto for a post-modern harmonic analysis, tailored to the
needs of mathematical signal processing. The proof uses in an essential way a
discretization of the problem by means of Gabor frames. We also show the
equivalence of the operator norm and the modulation space norm of the
corresponding kernel. For operators acting on $M^{p,q}$ a similar
characterization is not expected, but sufficient conditions for boundedness can
be sated in the same spirit.
