Diagonal coinvariants: old and new

The study of invariants and coinvariants is a classical topic that goes back to the early days of modern algebra. About 25 year ago, Garsia and Haiman initiated the study of diagonal coinvariants of the symmetric group in relation to what has become known as the "n! conjecture". In particular, the combinatorics behind these objects has been intensively studied in recent years, in connection with certain operators defined via the Macdonald symmetric functions. In this talk we review what is known on this subject, and present some of the surprising conjectures that arose in the last two years.