  This review gives an introduction to cohomological Donaldson-Thomas theory:
the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau
threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their
numerical DT invariant. Local and global aspects of the theory are both
covered, including representations of quivers with potential. We will discuss
the construction of the DT sheaf, a nontrivial topological coefficient system
on such a moduli space, along with some cohomology computations. The
Cohomological Hall Algebra, an algebra structure on cohomological DT spaces,
will also be introduced. The review closes with some recent appearances, and
extensions, of the cohomological DT story in the theory of knot invariants, of
cluster algebras, and elsewhere.
