  The twisted equivariant K-theory given by Freed and Moore is a K-theory which
unifies twisted equivariant complex K-theory, Atiyah's `Real' K-theory, and
their variants. In a general setting, we formulate this K-theory by using
Fredholm operators, and establish basic properties such as the Bott periodicity
and the Thom isomorphism. We also provide formulations of the K-theory based on
Karoubi's gradations in both infinite and finite dimensions, clarifying their
relationship with the Fredholm formulation.
