  This paper sets out basic properties of motivic twisted K-theory with respect
to degree three motivic cohomology classes of weight one. Motivic twisted
K-theory is defined in terms of such motivic cohomology classes by taking
pullbacks along the universal principal BG_m-bundle for the classifying space
of the multiplicative group scheme. We show a Kuenneth isomorphism for
homological motivic twisted K-groups computing the latter as a tensor product
of K-groups over the K-theory of BG_m. The proof employs an Adams Hopf
algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theory.
By adopting the notion of an E-infinity ring spectrum to the motivic homotopy
theoretic setting, we construct spectral sequences relating motivic
(co)homology groups to twisted K-groups. It generalizes various spectral
sequences computing the algebraic K-groups of schemes over fields. Moreover, we
construct a Chern character between motivic twisted K-theory and twisted
periodized rational motivic cohomology, and show that it is a rational
isomorphism. The paper includes a discussion of some open problems.
