  Fluctuations of global additive quantities, like total energy or
magnetization for instance, can in principle be described by statistics of sums
of (possibly correlated) random variables. Yet, it turns out that extreme
values (the largest value among a set of random variables) may also play a role
in the statistics of global quantities, in a direct or indirect way. This
review discusses different connections that may appear between problems of sums
and of extreme values of random variables, and emphasizes physical situations
in which such connections are relevant. Along this line of thought, standard
convergence theorems for sums and extreme values of independent and identically
distributed random variables are recalled, and some rigorous results as well as
more heuristic reasonings are presented for correlated or non-identically
distributed random variables. More specifically, the role of extreme values
within sums of broadly distributed variables is addressed, and a general
mapping between extreme values and sums is presented, allowing us to identify a
class of correlated random variables whose sum follows (generalized) extreme
value distributions. Possible applications of this specific class of random
variables are illustrated on the example of two simple physical models. A few
extensions to other related classes of random variables sharing similar
qualitative properties are also briefly discussed, in connection with the
so-called BHP distribution.
