  This is a survey on the intermittent behavior of the parabolic {Anderson}
model, which is the Cauchy problem for the heat equation with random potential
on the lattice $\Z^d$. We first introduce the model and give heuristic
explanations of the long-time behavior of the solution, both in the annealed
and the quenched setting for time-independent potentials. We thereby consider
examples of potentials studied in the literature. In the particularly important
case of an i.i.d. potential with double-exponential tails we formulate the
asymptotic results in detail. Furthermore, we explain that, under mild
regularity assumptions, there are only four different universality classes of
asymptotic behaviors. Finally, we study the moment Lyapunov exponents for
space-time homogeneous catalytic potentials generated by a {Poisson} field of
random walks.
