  This paper is concerned with the behaviour of a L\'{e}vy process when it
crosses over a positive level, $u$, starting from 0, both as $u$ becomes large
and as $u$ becomes small. Our main focus is on the time, $\tau_u$, it takes the
process to transit above the level, and in particular, on the {\it stability}
of this passage time; thus, essentially, whether or not $\tau_u$ behaves
linearly as $u\dto 0$ or $u\to\infty$. We also consider conditional stability
of $\tau_u$ when the process drifts to $-\infty$, a.s. This provides
information relevant to quantities associated with the ruin of an insurance
risk process, which we analyse under a Cram\'er condition.
