One of the most serious shortcomings of covariance/correlation are the assumptions of linearity and normality.
What is the most natural generalization of these measures of dependence when you want to model the dependence structure of extreme events using heavy-tailed distributions, e.g. the Generalized extreme value distribution?
With "most natural generalization" I mean that the classical covariance/correlation is included as a special case when the usual assumptions hold.
(Disclosure: After having received no answers for nearly two weeks I posted this question also at Quantitative Finance)