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This article contains proofs of the results announced in [21] in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one correspondences between orientations, Chern structures and Thom structures on a given ring cohomology theory. The theory is illustrated by motivic cohomology, algebraic K-theory, algebraic cobordism theory and by other examples.

Using the theory of framed correspondences developed by Voevodsky [22], we introduce and study framed motives of algebraic varieties. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension P 1 -spectrum of any smooth scheme X ∈ Sm/k. Moreover, it is shown that the bispectrumeach term of which is a twisted framed motive of X, has motivic homotopy type of the suspension bispectrum of X. Furthermore, an explicit computation of infinite P 1 -loop motivic spaces is given in terms of spaces with framed correspondences. We also introduce big framed motives of bispectra and show that they convert the classical Morel-Voevodsky motivic stable homotopy theory into an equivalent local theory of framed bispectra. As a topological application, it is proved that the framed motive M f r (pt)(pt) of the point pt = Spec k evaluated at pt is a quasifibrant model of the classical sphere spectrum whenever the base field k is algebraically closed of characteristic zero.

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Under a certain normalization assumption we prove that the P 1 -spectrum BGL of Voevodsky which represents algebraic K-theory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1 -spectrum BGL with the structure of a commutative P 1 -ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec(Z). For an arbitrary Noetherian scheme S of finite Krull dimension we pull this structure back to obtain a distinguished monoidal structure on BGL. This monoidal structure is relevant for our proof of the motivic Conner-Floyd theorem [PPR]. It has also been used to obtain a motivic version of Snaith's theorem [GS].This paper is concerned with results in motivic homotopy theory, which was put on firm foundations by Morel and Voevodsky in [MV] and [V]. Due to technical reasons explained below, the setup in [MV], as well as other model categories used in motivic homotopy theory, are inconvenient for our purposes, so we decided to pursue a slightly different approach. We refer to the Appendix A for the basic terminology, notation, constructions, definitions, and results concerning motivic homotopy theory. For a Noetherian scheme S of finite Krull dimension we write M(S), M • (S), H • (S) and SH(S) for the category of motivic spaces, the category of pointed motivic spaces, the pointed motivic homotopy