  We study the extremes of multivariate regularly varying random fields. The
crucial tools in our study are the tail field and the spectral field, notions
that extend the tail and spectral processes of Basrak and Segers (2009). The
spatial context requires multiple notions of extremal index, and the tail and
spectral fields are applied to clarify these notions and other aspects of
extremal clusters. An important application of the techniques we develop is to
the Brown-Resnick random fields.
