group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a space equipped with a notion of dimension $dim X \in \mathbb{N}$ and a notion of Kähler differential forms, a $\Theta$-characteristic of $X$ is a choice of square root of the canonical characteristic class of $X$. See there for more details.
In complex analytic geometry and at least if the Theta characteristic is principally polarizing then its holomorphic sections are called theta functions.
For $\Sigma$ a Riemann surface, the choices of square roots of the canonical bundle correspond to the choice of spin structures.
For $X$ of genus $g$, there are $2^{2g}$ many choices of square roots of the canonical bundle.
The first statement remains true in higher dimensions over Kähler manifolds, see at Spin structure – On Kähler manifolds.