de Rham cohomology in various settings including Algebraic de Rham cohomology, and the fact that over Z we get abelian groups rather than vector spaces.
Not sure if this belongs here: Derived de Rham cohomology
Generalized de Rham cohomology
Cohomology of currents. There is a ref to Griffiths-Harris, and also some reading notes which can be found under Deligne cohomology (I think)
Dolbeault cohomology
nLab page on E25 Notions of de Rham cohomology