(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A circle bundle is a principal bundle for the circle group $S^1$. Equivalently this is a $U(1)$-principal bundle, for the unitary group $U(1)$.
Under the canonical representation $\mathbf{B}U(1) \to Vect_{\mathbb{C}}$ the corresponding associated bundle is a complex line bundle.