group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The generalized homology theory which is represented by the sphere spectrum $\mathbb{S}$ is usually called stable homotopy, since the homology groups that it assigns to a suitable topological space $X$ are just the stable homotopy group of $X$:
Hence the coefficient cohomology ring of stable homotopy homology theory (its value on the point) is the stable homotopy groups of spheres. This highlights that stable homotopy homology of any space $X$ is extremely hard, or impossible, to completely analyze, since this is true already for the coefficient ring over the point.
Beware that the term stable homotopy theory, which would seem to be the canonical name for this generalized homology theory, traditionally refers instead to the general homotopy theory of spectra. The full term stable homotopy homology theory is used for emphasis, but clunky in practice.
The dual generalized cohomology theory, co-represented by the sphere spectrum, is called stable cohomotopy theory.
Along the canonical morphism of spectra $\mathbb{S} \to H R$ from the sphere spectrum to any Eilenberg-MacLane spectrum of a ring $R$ (which is the unit map of $H R$ in the (infinity,1)-category of E-infinity ring spectra) stable homotopy homology maps to ordinary homology with coefficients in $R$ (given notably by singular homology).
This is known as the Hurewicz homomorphism
More generally, for $E$ an E-infinity ring spectrum, the unit $\mathbb{S} \to E$ induces a natural transformation from stable homotopy homology theory to the generalized homology theory co-represented by $E$.
This is known as the Boardman homomorphism.