  In the 1960's, four famous scaling relations were developed which relate the
six standard critical exponents describing continuous phase transitions in the
thermodynamic limit of statistical physics models. They are well understood at
a fundamental level through the renormalization group. They have been verified
in multitudes of theoretical, computational and experimental studies and are
firmly established and profoundly important for our understanding of critical
phenomena. One of the scaling relations, hyperscaling, fails above the upper
critical dimension. There, critical phenomena are governed by Gaussian fixed
points in the renormalization-group formalism. Dangerous irrelevant variables
are required to deliver the mean-field and Landau values of the critical
exponents, which are deemed valid by the Ginzburg criterion. Also above the
upper critical dimension, the standard picture is that, unlike for
low-dimensional systems, finite-size scaling is non-universal. Here we report
on new developments which indicate that the current paradigm is flawed and
incomplete. In particular, the introduction of a new exponent characterising
the finite-size correlation length allows one to extend hyperscaling beyond the
upper critical dimension. Moreover, finite-size scaling is shown to be
universal provided the correct scaling window is chosen. These recent
developments also lead to the introduction of a new scaling relation analogous
to one introduced by Fisher 50 years ago.
