group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
The Grothendieck-Riemann-Roch theorem describes (the failure of) the naturality of the behaviour of a Chern character under push forward along proper maps.
It says, in the formulation of (Atiyah-Hirzebruch 61), that for $f \colon X \longrightarrow Y$ a flat morphism of schemes which are flat and regular quasi-projective varieties over the spectrum of a Dedekind domain, then the Chern character of the push-forward of some $E$ is the push-forward of the cup product of the Chern-character of $E$ with the Todd class. Hence it says that “Chern-cup-Todd is natural under pushforward” along proper maps, and generally along K-oriented maps.
If $f \colon X \to Y$ is a proper map, then there is a commuting diagram
where…
If $X$ is an algebraic curve, then the Riemann-Roch theorem reduces to a statement about the Euler characteristic/curve. This generalizes to arithmetic geometry with the notion of genus of a number field.
The formulation in terms of topological K-theory is due to
For a general survey see
Wikipedia, Grothendieck-Riemann-Roch theorem
Wikipedia, Riemann-Roch theorem for surfaces
Discussion of Riemann-Roch over arithmetic curves is in
The refinement to differential cohomology, hence differential K-theory, is discussed in section 6.2 of
Over algebraic stacks/Deligne-Mumford stacks the GRR theorem is discussed in
Roy Joshua, Riemann-Roch for algebraic stacks (pdf I, pdf II, pdf III)
Bertrand Toën, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory 18 (1999), no. 1, 33–76. 1, 23
Dan Edidin, Riemann-Roch for Deligne-Mumford stacks (arXiv:1205.4742v1)