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

Passage time and fluctuation calculations for subexponential L\'evy processes