The multivariate extremal index has been introduced by Nandagopalan as a measure of the clustering among the extreme values of a multivariate stationary process. In this paper, we derive some additional properties and use those to construct a statistical estimation scheme. Central to the discussion is a class of processes we call M4 processes, which are characterized by means of a multivariate generalization of a characterization due to Deheuvels for univariate max-stable processes. The multivariate extremal index for M4 processes is derived, and certain properties established. We then discuss estimation of the multivariate extremal index. This is based on joint work with Richard L. Smith.