  Nonuniform ellipticity is a classical topic in the theory of partial
differential equations. While several results in regularity theory have been
adding up over decades, many basic issues, as for instance the validity of
Schauder theory and sharp dependence of regularity upon data, remained opened
for a while. In these notes we give an overview of recent results and
techniques about the topic, that, via a novel use of nonlinear potential
theoretic methods, allow to answer several of the above questions.
