The idea of
twisted
K
-theory
can be traced back to the paper
[
DK70
]
by Donovan and Karoubi. Its recent popularity is
mainly due to the appearance of twisted
K
-theory in
phypsics
.

We also have twisted versions of generalized
cohomology theories other than
K
-theory
. The cohomology with
local coefficients
can be also regarded as a kind of twisted cohomology
theory.

Atiyah and Segal brieﬂy describe an idea of
twisting
generalized cohomology theories
in
[
AS04
]
. And they discuss twistings of de Rham cohomology in
§
6 of
[
AS06
]
.

There is an attempt of axiomatization of twisted
generalized cohomology by Bunke and Schick
[
BSa
]
. They also discuss axioms of twisted cohomology
theory for
orbispaces
in
[
BSb
]
.

Although there is a description of how to construct a
twisted version of a generalized cohomology theory in
the above mentioned paper by Atiyah and Segal, we
should probably use parametrized spectra as is done by
Waldmüller in
[
Wal
]
. He constructs spectra representing
twisted
K
-theory
and twisted
Spin
^{
c
}
-cobordism. See Part V of the book
[
MS06
]
by May and Sigurdsson, for more details. There is
also a description of equivariant versions.

Ando, Blumberg, and Gepner
[
ABG
^{
+
}
b
,
ABGa
]
consider twistings as Thom spectra (in a general
sense).

Sati
[
Sat
]
suggests to use twistings of
tmf
for
M-theory
.

[ABGa]
Matthew Ando, Andrew J. Blumberg, and David
Gepner. Twists of
K
-theory and
TMF
,
arXiv:1002.3004
.

[ABG
^{
+
}
b]
Matthew Ando, Andrew J. Blumberg,
David J. Gepner, Michael J. Hopkins,
and Charles Rezk. Units of ring spectra and Thom
spectra,
arXiv:0810.4535
.

[AS04]
Michael Atiyah and Graeme Segal. Twisted
K
-theory.
Ukr. Mat.
Visn.
, 1(3):287–330, 2004,
arXiv:math.KT/0407054
.

[AS06]
Michael Atiyah and Graeme Segal. Twisted
K
-theory and cohomology. In
Inspired by S. S. Chern
, volume 11 of
Nankai
Tracts Math.
, pages 5–43. World Sci. Publ., Hackensack,
NJ, 2006,
arXiv:math.KT/0510674
.

[BSa]
Ulrich Bunke and Thomas Schick. On the topology of
T
-duality,
arXiv:math.GT/0405132
.

[BSb]
Ulrich Bunke and Thomas Schick.
T
-duality for non-free circle actions,
arXiv:math.GT/0508550
.

[DK70]
P. Donovan and M. Karoubi. Graded
Brauer groups and
K
-theory with local coefficients.
Inst. Hautes
Études Sci. Publ. Math.
, (38):5–25, 1970.

[MS06]
J. P. May and J. Sigurdsson.
Parametrized homotopy theory
, volume 132 of
Mathematical Surveys and Monographs
. American Mathematical Society, Providence, RI,
2006.

[Sat]
Hisham Sati. Geometric and topological structures
related to M-branes,
arXiv:1001.5020
.

[Wal]
Robert Waldmüller. Products and push-forwards
in parametrised cohomology theories,
arXiv:math.AT/0611225
.