The aim of this lecture note is to survey recent developments in cohomological Donaldson–Thomas (CoDT) theory, which is a categorification of the numerical Donaldson–Thomas theory counting coherent sheaves on 3-Calabi–Yau varieties. One of the main goals is to state Joyce’s conjecture about a functorial behavior of the DT perverse sheaves, which has many consequences such as the existence of an algebra structure on the CoDT invariants. We also discuss recent applications of the CoDT theory to the cohomological study of the moduli space of Higgs bundles on Riemann surfaces.