group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Dolbeault cohomology of a complex manifold $X$ is the abelian sheaf cohomology $H^q(X;\Omega_X^p)$, of the abelian sheaf $\Omega_X^p$ is Dolbeault complex of holomorphic $p$-forms.
Given any holomorphic vector bundle $E$, one can form the Dolbeault resolution $E \otimes \Omega^{0,q}$, where $\Omega^{0,q}$ is the sheaf of $C^\infty$ $(0,q)$-forms. This is an acyclic resolution of $E$ and hence computes its sheaf cohomology.