  The probabilistic characterization of the relationship between two or more
random variables calls for a notion of dependence. Dependence modeling leads to
mathematical and statistical challenges, and recent developments in extremal
dependence concepts have drawn a lot of attention to probability and its
applications in several disciplines. The aim of this paper is to review various
concepts of extremal positive and negative dependence, including several
recently established results, reconstruct their history, link them to
probabilistic optimization problems, and provide a list of open questions in
this area. While the concept of extremal positive dependence is agreed upon for
random vectors of arbitrary dimensions, various notions of extremal negative
dependence arise when more than two random variables are involved. We review
existing popular concepts of extremal negative dependence given in literature
and introduce a novel notion, which in a general sense includes the existing
ones as particular cases. Even if much of the literature on dependence is
focused on positive dependence, we show that negative dependence plays an
equally important role in the solution of many optimization problems. While the
most popular tool used nowadays to model dependence is that of a copula
function, in this paper we use the equivalent concept of a set of
rearrangements. This is not only for historical reasons. Rearrangement
functions describe the relationship between random variables in a completely
deterministic way, allow a deeper understanding of dependence itself, and have
several advantages on the approximation of solutions in a broad class of
optimization problems.
