group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The de Rham complex $\Omega^\bullet(X)$ of a space $X$ is the cochain complex that in degree $n$ has the differential forms (which may mean: Kähler differential forms) of degree $n$, and whose differential is the de Rham differential or exterior derivative.
(named after Georges de Rham)
The de Rham complex of a smooth manifold is the cochain complex which in degree $n \in \mathbb{N}$ has the vector space $\Omega^n(X)$ of degree-$n$ differential forms on $X$. The coboundary map is the deRham exterior derivative. The cohomology of the de Rham complex is de Rham cohomology.
Under the wedge product, the deRham complex becomes a differential graded algebra. This may be regarded as the Chevalley–Eilenberg algebra of the tangent Lie algebroid $T X$ of $X$.
For smooth varieties $X$, algebraic de Rham cohomology is defined to be the hypercohomology of the de Rham complex $\Omega_X^\bullet$.
De Rham cohomology has a rather subtle generalization for possibly singular algebraic varieties due to (Grothendieck).
For analytic spaces
Hodge-de Rham spectral sequence?
crystalline cohomology, comparison theorem (crystalline cohomology)
(…)
A useful introduction is
A classical reference on the algebraic version is
See also
Yves André, Comparison theorems between algebraic and analytic De Rham cohomology (pdf)
Mikhail Kapranov, DG-Modules and vanishing cycles (KapranovDGModuleVanishingCycle.pdf?)