  The notion of multivariate upcrossings index of a stationary sequence
${\bf{X}}=\{(X_{n,1},\ldots,X_{n,d})\}_{n\geq 1}$ is introduced and its main
properties are derived, namely the relations with the multivariate extremal
index and the clustering of upcrossings. Under asymptotic independence
conditions on the marginal sequences of ${\bf{X}}$ the multivariate upcrossings
index is obtained from the marginal upcrossings indices. For a class of
stationary multidimensional sequences assumed to satisfy a mild oscillation
restriction, the multivariate upcrossings index is computed from the joint
distribution of a finite number of variables. The upcrossings index is
calculated for two examples of bivariate sequences.
