We develop an operator algebraic model for twisted K-theory, which includes the most general twistings as a generalized cohomology theory (that is, all those classified by the unit spectrum bgl1(KU)). Our model is based on strongly self-absorbing C*-algebras. We compare it with the known homotopy-theoretic descriptions in the literature, which either use parametrized stable homotopy theory or 8-categories. We derive a similar comparison of analytic twisted K-homology with its topological counterpart based on generalized Thom spectra. Our model also works for twisted versions of localizations of the K-theory spectrum, like KU[1] or KUQ.
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