Certain three-dimensional images arising in medicine and astrophysics are modelled as a smooth random field, and experimenters are interested in the number of peaks or "hot-spots" present in such an image. This paper studies the Hadwiger characteristic of the excursion set of a random field. The excursion set is the set of points where the image exceeds a fixed threshold, and the Hadwiger characteristic, like the Euler characteristic, counts the number of connected components in the excursion set minus the number of "holes." For high thresholds the Hadwiger characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler, who defined the IG (integral geometry) characteristic of excursion sets as a multidimensional analogue of the number of "upcrossings" of the threshold by a unidimensional process. The IG characteristic equals the Euler characteristic of an excursion set provided that the set does not touch the boundary of the volume, and Adler found the expected IG characteristic for a stationary random field inside a fixed volume. Worsley, Evans, Marrett and Neelin used the IG characteristic as an estimator of the number of regions of activation of positron emission tomography (PET) images of blood flow in the brain. Unfortunately the IG characteristic is only defined on intervals: it is not invariant under rotations and it can fail to count connected regions that touch the boundary. This is important since activation often occurs in the cortical regions near the boundary of the brain. In this paper we study the Hadwiger characteristic, which is defined on arbitrary sets, is invariant under rotations and does count connected regions whether they touch the boundary or not. Our main result is a simple expression for the expected Hadwiger characteristic for an isotropic stationary random field in two and three dimensions and on a smooth surface embedded in three dimensions. Results are applied to PET studies of pain perception and word recognition.