  We study the parabolic Anderson problem, that is, the heat equation
$\partial_tu=\Delta u+\xi u$ on $(0,\infty)\times{\mathbb{Z}}^d$ with
independent identically distributed random potential
$\{\xi(z):z\in{\mathbb{Z}}^d\}$ and localized initial condition
$u(0,x)={\mathbf{1}}_0(x)$. Our interest is in the long-term behavior of the
random total mass $U(t)=\sum_zu(t,z)$ of the unique nonnegative solution in the
case that the distribution of $\xi(0)$ is heavy tailed. For this, we study two
paradigm cases of distributions with infinite moment generating functions: the
case of polynomial or Pareto tails, and the case of stretched exponential or
Weibull tails. In both cases we find asymptotic expansions for the logarithm of
the total mass up to the first random term, which we describe in terms of weak
limit theorems. In the case of polynomial tails, already the leading term in
the expansion is random. For stretched exponential tails, we observe random
fluctuations in the almost sure asymptotics of the second term of the
expansion, but in the weak sense the fourth term is the first random term of
the expansion. The main tool in our proofs is extreme value theory.
