  These lecture notes provide an overview of the renormalization group (RG) as
a successful framework to understand critical phenomena above the upper
critical dimension $d_{\rm uc}$. After an introduction to the scaling picture
of continuous phase transitions, we discuss the apparent failure of the
Gaussian fixed point to capture scaling for Landau mean-field theory, which
should hold in the thermodynamic limit above $d_{\rm uc}$. We recount how
Fisher's dangerous-irrelevant-variable formalism applied to thermodynamic
functions partially repairs the situation but at the expense of hyperscaling
and finite-size scaling, both of which were, until recently, believed not to
apply above $d_{\rm uc}$. We recall limitations of various attempts to match
the RG with analytical and numerical results for Ising systems. We explain how
the extension of dangerous irrelevancy to the correlation sector is key to
marrying the above concepts into a comprehensive RG scaling picture that
renders hyperscaling and finite-size scaling valid in all dimensions. We
collect what we believe is the current status of the theory, including some new
insights and results. This paper is in grateful memory of Michael Fisher who
introduced many of the concepts discussed and who, half a century later,
contributed to their advancement.
