Abstract: The problem of describing the behaviour of a random walk or Levy process which drifts to -infinity, conditional on exceeding a high barrier, has received a lot of attention, often in the guise of “the ruin problem”. However this has generally been in the case of a finite mean. In this talk I will survey the known results in the finite mean case, and present some new results for the infinite mean case. These results have been developed in conjunction with Ross Maller (ANU) and Claudia Kluppelberg (Munich).