AbstractWe make some computations in stable motivic homotopy theory over Spec ℂ, completed at 2. Using homotopy fixed points and the algebraicK-theory spectrum, we construct over ℂ a motivic analogue of the realK-theory spectrumKO. We also establish a theory of motivic connective covers over ℂ to obtain a motivic version ofko. We establish an Adams spectral sequence for computing motivicko-homology. TheE2-term of this spectral sequence involves Ext groups over the subalgebraA(1) of the motivic Steenrod algebra. We make several explicit computations of theseE2-terms in interesting special cases.