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, τu, 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 τu behaves
linearly as u\dto 0 or u→∞. We also consider conditional stability
of τu when the process drifts to −∞, 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