  We introduce quantum tomography on locally compact Abelian groups $G$. A
linear map from the set of quantum states on the $C^*$-algebra $A(G)$ generated
by the projective unitary representation of $G$ to the space of characteristic
functions is constructed. The dual map determining symbols of quantum
observables from $A(G)$ is derived. Given a characteristic function of a state
the quantum tomogram consisting a set of probability distributions is
introduced. We provide three examples in which $G={\mathbb R}$ (the optical
tomography), $G={\mathbb Z}_n$ (corresponding to measurements in mutually
unbiased bases) and $G={\mathbb T}$ (the tomography of the phase). As an
application we have calculated the quantum tomogram for the output states of
quantum Weyl channels.
