  We consider the passage time problem for L\'evy processes, emphasising heavy
tailed cases. Results are obtained under quite mild assumptions, namely, drift
to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean
case), but possibly much faster (the infinite mean case), together with
subexponential growth on the positive side. Local and functional versions of
limit distributions are derived for the passage time itself, as well as for the
position of the process just prior to passage, and the overshoot of a high
level. A significant connection is made with extreme value theory via regular
variation or maximum domain of attraction conditions imposed on the positive
tail of the canonical measure, which are shown to be necessary for the kind of
convergence behaviour we are interested in.
