  We investigate the moment asymptotics of the solution to the stochastic heat
equation driven by a $(d+1)$-dimensional L\'evy space--time white noise. Unlike
the case of Gaussian noise, the solution typically has no finite moments of
order $1+2/d$ or higher. Intermittency of order $p$, that is, the exponential
growth of the $p$th moment as time tends to infinity, is established in
dimension $d=1$ for all values $p\in(1,3)$, and in higher dimensions for some
$p\in(1,1+2/d)$. The proof relies on a new moment lower bound for stochastic
integrals against compensated Poisson measures. The behavior of the
intermittency exponents when $p\to 1+2/d$ further indicates that intermittency
in the presence of jumps is much stronger than in equations with Gaussian
noise. The effect of other parameters like the diffusion constant or the noise
intensity on intermittency will also be analyzed in detail.
