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.
For $X$ a Riemann surface, the choices of square roots of the canonical bundle correspond to the choice of spin structures (Atiyah, prop. 3.2). For $X$ of genus $g$, there are $2^{2g}$ many choices of square roots of the canonical bundle.
The function that sends a square root line bundle to the dimension of its space of holomorphic sections $mod 2$ is a quadratic refinement of the intersection pairing on $H^1(X, \mathbb{Z}_2)$ (Atiyah, theorem 2).
The following table lists classes of examples of square roots of line bundles
The spaces of choice of $\Theta$-characteristics over Riemannian manifolds were discussed in
See also
Gavril Farkas, Theta characteristics and their moduli (2012) (arXiv:1201.2557)
M. Bertola, Riemann surfaces and Theta Functions, August 2010 (pdf)