group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A differential form $\omega \in \Omega^n(X)$ on a Riemannian manifold $(X,g)$ is called a harmonic fom if it is in the kernel of the Laplace operator $\Delta_g$ of $X$ in that $\Delta \omega = (d + d^\dagger)^2 \omega = 0$.
The basic properties of harmonic forms are described by Hodge theory. See there for details.