HOD is the class of hereditarily ordinal-definable sets. There is an ongoing conversation in contemporary set theory concerning how far HOD can diverge from V, particularly in connection with Woodin’s HOD Hypothesis and the HOD Conjecture.
We will present several definitions and ideas needed to understand these questions. We briefly survey some of the directions in which these divergences have been studied and then focus on the extent to which large cardinal properties are transferred into HOD, particularly in the region between supercompact and extendible cardinals. Since HOD is \Sigma_2 definable, it is also natural to consider variants of supercompactness that incorporate forms of \Sigma_2-correctness and to consider how these notions interact with HOD.