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Fluctuations of global additive quantities, like total energy ormagnetization for instance, can in principle be described by statistics of sumsof (possibly correlated) random variables. Yet, it turns out that extremevalues (the largest value among a set of random variables) may also play a rolein the statistics of global quantities, in a direct or indirect way. Thisreview discusses different connections that may appear between problems of sumsand of extreme values of random variables, and emphasizes physical situationsin which such connections are relevant. Along this line of thought, standardconvergence theorems for sums and extreme values of independent and identicallydistributed random variables are recalled, and some rigorous results as well asmore heuristic reasonings are presented for correlated or non-identicallydistributed random variables. More specifically, the role of extreme valueswithin sums of broadly distributed variables is addressed, and a generalmapping between extreme values and sums is presented, allowing us to identify aclass of correlated random variables whose sum follows (generalized) extremevalue distributions. Possible applications of this specific class of randomvariables are illustrated on the example of two simple physical models. A fewextensions to other related classes of random variables sharing similarqualitative properties are also briefly discussed, in connection with theso-called BHP distribution.

Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics