If D is the derived category of a ring or space, then there is a natural ring homomorphism from Hochschild cohomology to the graded centre of that category, the Hochschild-Atiyah-Chern character. This homomorphism is yet poorly understood and I will discuss what we know and what we want to know, as well as ingredients, such as residues and comparison of Hochschild and Hochschild-Mitchell cohomology, that may help to further our understanding. I will also discuss the stabilized version of this homomorphism into the singularity category of a Gorenstein singularity.