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This is joint work with G .Garkusha. Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives is introduced and studied. To any smooth algebraic varietyÌýX, the framed motiveÌýMfr(X)Ìýis associated in that category. Theorem. The bispectrumÌý(MfrX,Mfr(X)(1),Mfr(X)(2),...),Ìýeach term of which is a twisted framed motive ofÌýX, has motivic homotopy type of the suspension bispectrum ofÌýX. (this result is anÌýA1-homotopy analog of a theorem due to G.Segal). We also construct a compactly generated triangulated category of framed bispectra and show that it reconstructs the Morel--Voevodsky categoryÌýSH(k). This machinery allows to recover in characteristic zero the celebrated theorem due to F. Morel stating that the stableÌýÏ€0,0(k)=Ìýthe Grothendiek-Witt ring of the fieldÌýkÌý. Also this machinery makes approachable Serre finitness conjecture: rationalÌýÏ€n,0(the reals)=0ÌýifÌýnÌýis not zero.Ìý
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