The issue of regularity has played a central role in the theory of
Partial Differential Equations almost since its inception, and despite
the tremendous advances made it still remains a very fruitful research
field. In particular considerable strides have been made in regularity
estimates for degenerate and singular elliptic and parabolic equations
over the last several years, and in many unexpected and challenging
directions. Because of all these recent results, it seemed high time to
create an overview that would highlight emerging trends and issues in
this fascinating research topic in a proper and effective way. The
course aimed to show the deep connections between these topics and to
open new research directions through the contributions of leading
experts in all of these fields.