We consider a continuous, infinitely divisible random field in Rd given as an
integral of a kernel function with respect to a Lévy basis with convolution
equivalent Lévy measure. For a large class of such random fields we compute
the asymptotic probability that the supremum of the field exceeds the level x
as x ! 1. Our main result is that the asymptotic probability is equivalent to
the right tail of the underlying Lévy measure.