The study of the classical theta operator was key to Edixhoven's proof of the weight part of Serre's modularity conjecture. Since then, a lot of work has been devoted to extending the construction of this operator to other Shimura varieties, with an eye towards generalisations of Serre's conjecture.
My goal is to give an overview of a family of generalised theta operators, on certain unitary Shimura varieties, that I constructed in my thesis and studied in subsequent work.