  The extremal index is a quantity introduced in extreme value theory to
measure the presence of clusters of exceedances. In the dynamical systems
framework, it provides important information about the dynamics of the
underlying systems. In this paper we provide a review of the meaning of the
extremal index in dynamical systems. Depending on the observables used, this
quantity can inform on local properties of attractors such as periodicity,
stability and persistence in phase space, or on global properties such as the
Lyapunov exponents. We also introduce a new estimator of the extremal index and
shows its relation with those previously introduced in the statistical
literature. We reserve a particular focus to the systems perturbed with noise
as they are a good paradigm of many natural phenomena. Different kind of noises
are investigated in the annealed and quenched situations. Applications to
climate data are also presented.
