  We prove general kernel theorems for operators acting between coorbit spaces.
These are Banach spaces associated to an integrable representation of a locally
compact group and contain most of the usual function spaces (Besov spaces,
modulation spaces, etc.). A kernel theorem describes the form of every bounded
operator between a coorbit space of test functions and distributions by means
of a kernel in a coorbit space associated to the tensor product representation.
As special cases we recover Feichtinger's kernel theorem for modulation spaces
and the recent generalizations by Cordero and Nicola. We also obtain a kernel
theorem for operators between the Besov spaces $\dot{B}^0_{1,1}$ and
$\dot{B}^{0}_{\infty, \infty }$.
