We study the stochastic heat equation (SHE) $\partial_t u = \frac12 Δu + βu ξ$ driven by a multiplicative Lévy noise $ξ$ with positive jumps and amplitude $β>0$, in arbitrary dimension $d\geq 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\geq3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple Lévy integrals arising in the chaos decomposition of $u$, we further show that the solution has finite $p$th moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order $p$ of the solution, which are asymptotically sharp in the limit as $β\to0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial Lévy measure, at any disorder intensity $β>0$, in any dimension $d\geq1$.