group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
What is usually called “generalized cohomology” is obtained from a generalization of the axioms of Eilenberg–Steenrod.
By Brown's representability theorem this is equivalent to cohomology with coefficient object a spectrum. See
Notice that there are “generalizations of generalized cohomology”. See the discussion at cohomology.