BPS cohomology is a cohomology theory that ``categorifies" refined BPS invariants associated to 3-Calabi-Yau categories, in the sense that this cohomology recovers these invariants after passing to the (virtual) Poincaré polynomial of the BPS cohomology. As well as categorifying refined BPS state counts, this cohomology turns out to have a rich algebraic structure, with links to various constructions and central objects in quantum algebra and geometric representation theory. I will survey the construction of BPS cohomology, as well as applications in the above areas and beyond.