The subject of this tutorial is Woodin's HOD conjecture, one of the most prominent open problems in pure set theory. We begin with a proof of his HOD dichotomy theorem along with an improvement of the speaker's reducing the large cardinal hypothesis from an extendible to a strongly compact cardinal. Following this, we mostly discuss the implications of the failure of the HOD conjecture, especially $\omega$-strongly measurable cardinals and a condition under which such cardinals are locally supercompact in $\HOD$.