  We re-consider Leadbetter's extremal index for stationary sequences. It has
interpretation as reciprocal of the expected size of an extremal cluster above
high thresholds. We focus on heavy-tailed time series, in particular on
regularly varying stationary sequences, and discuss recent research in extreme
value theory for these models. A regularly varying time series has multivariate
regularly varying finite-dimensional distributions. Thanks to results by Basrak
and Segers we have explicit representations of the limiting cluster structure
of extremes, leading to explicit expressions of the limiting point process of
exceedances and the extremal index as a summary measure of extremal clustering.
The extremal index appears in various situations which do not seem to be
directly related, like the convergence of maxima and point processes. We
consider different representations of the extremal index which arise from the
considered context. We discuss the theory and apply it to a regularly varying
AR(1) process and the solution to an affine stochastic recurrence equation
