  In the classical theorems of extreme value theory the limits of suitably
rescaled maxima of sequences of independent, identically distributed random
variables are studied. So far, only affine rescalings have been considered. We
show, however, that more general rescalings are natural and lead to new limit
distributions, apart from the Gumbel, Weibull, and Fr\'echet families. The
problem is approached using the language of Renormalization Group
transformations in the space of probability densities. The limit distributions
are fixed points of the transformation and the study of the differential around
them allows a local analysis of the domains of attraction and the computation
of finite-size corrections.
