  We consider last-passage percolation models in two dimensions, in which the
underlying weight distribution has a heavy tail of index alpha<2. We prove
scaling laws and asymptotic distributions, both for the passage times and for
the shape of optimal paths; these are expressed in terms of a family (indexed
by alpha) of "continuous last-passage percolation" models in the unit square.
In the extreme case alpha=0 (corresponding to a distribution with slowly
varying tail) the asymptotic distribution of the optimal path can be
represented by a random self-similar measure on [0,1], whose multifractal
spectrum we compute. By extending the continuous last-passage percolation model
to R^2 we obtain a heavy-tailed analogue of the Airy process, representing the
limit of appropriately scaled vectors of passage times to different points in
the plane. We give corresponding results for a directed percolation problem
based on alpha-stable Levy processes, and indicate extensions of the results to
higher dimensions.
