We consider smooth, infinitely divisible random fields with regularly varying Levy measure, and are interested in the geometric characteristics of the excursion sets over high levels u.

For a large class of such random fields we compute the asymptotic
joint distribution of the numbers of critical points, of various types, of the random field in the excursion set, conditional on the latter being non-empty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set.

In a significant departure from the Gaussian situation, the high level excursion sets for these random fields can have quite a complicated geometry. Whereas in the Gaussian case non-empty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.