  Let $L$ be a L\'evy operator. A function $h$ is said to be harmonic with
respect to $L$ if $L h = 0$ in an appropriate sense. We prove Liouville's
theorem for positive functions harmonic with respect to a general L\'evy
operator $L$: such functions are necessarily mixtures of exponentials. For
signed harmonic functions we provide a fairly general result, which encompasses
and extends all Liouville-type theorems previously known in this context, and
which allows to trade regularity assumptions on $L$ for growth restrictions on
$h$. Finally, we construct an explicit counterexample which shows that
Liouville's theorem for signed functions harmonic with respect to a general
L\'evy operator $L$ does not hold.
