symmetric monoidal (∞,1)-category of spectra
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Topological Hochschild (resp. topological cyclic) homology is a refinement of Hochschild homology/cyclic homology from commutative rings/algebras to the higher algebra of ring spectra/E-∞ rings/E-∞ algebras. A good survey is in (May).
In particular, given a ring $R$, then there is a natural morphism of spectra
from the algebraic K-theory spectrum to the topological Hochschild homology spectrum and factoring through the topological cyclic homology spectrum called the cyclotomic trace which much like a Chern character map for algebraic K-theory.
The originalreferences
Marcel Bökstedt, Topological Hochschild homology, Bielefeld, 1985, 1988
Marcel Bökstedt, W.C. Hsiang, Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 463-539, MR94g:55011, doi
Marcel Bökstedt, Ib Madsen, Topological cyclic homology of the integers, $K$-theory (Strasbourg, 1992). Astérisque 226 (1994), 7–8, 57–143.
and a further generalization is defined in
Review and exposition includes
Peter May, Topological Hochschild and Cyclic Homology and Algebraic K-theory (pdf)
Teena Gerhardt, Computations in algebraic K-theory, talk at CUNY Workshop on differential cohomologies 2014 (video recording)
Ricardo Andrade, THH notes, MIT juvitop seminar pdf, babytop seminar pdf
Anatoly Preygel, Hochschild homology notes, juvitop seminar, pdf
These theories are the target for the trace map from $K$-theory, so they can be viewed as an approximation to algebraic K-theory of a ring spectrum.
T. Pirashvili, F. Waldhausen, Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra 82 (1992), 81-98, MR96d:19005, doi
T. Pirashvili, On the topological Hochschild homology of $\mathbf{Z}/p^k\mathbf{Z}$, Comm. Algebra 23 (1995), no. 4, 1545–1549, MR97h:19007, doi
Z. Fiedorowicz, T. Pirashvili, R. Schwänzl, R. Vogt, F. Waldhausen, Mac Lane homology and topological Hochschild homology, Math. Ann. 303 (1995), no. 1, 149–164, MR97h:19007, doi
Bjørn Ian Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), 223-242, pdf
Thomas Geisser, Lars Hesselhoft, Topological cyclic homology of schemes, in: Algebraic $K$-theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math. 67, Amer. Math. Soc. 1999, MR2001g:19003; K-theory archive
Thomas Geisser, Motivic Cohomology, K-Theory and Topological Cyclic Homology, Handbook of K-theory II.1, pdf
R. McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222.
Gunnar Carlsson, Christopher L. Douglas, Bjørn Ian Dundas, Higher topological cyclic homology and the Segal conjecture for tori, Adv. Math. 226 (2011), no. 2, 1823–1874, MR2737802, doi
Ib Madsen, Algebraic K-theory and traces, pdf
Dundas, Goodwillie and McCarthy, The local structure of algebraic K-theory, book in progress, pdf
J. McClure, R. Staffeldt, On the topological Hochschild homology of $b u$, I, pdf
Daniel Joseph Vera, Topological Hochschild homology of twisted group algebra, MIT Ph. D. thesis 2006, pdf