  Studying the geometry generated by Gaussian and Gaussian- related random
fields via their excursion sets is now a well developed and well understood
subject. The purely non-Gaussian scenario has, however, not been studied at
all. In this paper we look at three classes of stable random fields, and obtain
asymptotic formulae for the mean values of various geometric characteristics of
their excursion sets over high levels.
  While the formulae are asymptotic, they contain enough information to show
that not only do stable random fields exhibit geometric behaviour very
different from that of Gaussian fields, but they also differ significantly
among themselves.
