  In this paper we deal with the classical problem of random cover times. We
investigate the distribution of the time it takes for a Poisson process of
cylinders to cover a set $A \subset \mathbb{R}^d.$ This Poisson process of
cylinders is invariant under rotations, reflections and translations, and in
addition we add a time component so that cylinders are "raining from the sky"
at unit rate. Our main results concerns the asymptotic of this cover time as
the set $A$ grows. If the set $A$ is discrete and well separated, we show
convergence of the cover time to a Gumbel distribution. If instead $A$ has
positive box dimension (and satisfies a weak additional assumption), we find
the correct rate of convergence.
