We generalize Rost's theory of cycle modules ([25]) using Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a (quadratic) setting to study general cycle complexes and their (co)homology groups. The usual constructions are developped: proper pushfoward, (essentially) smooth pullback, long exact sequences, (coniveau) spectral sequences, homotopy invariance and products. Moreover, we define Gysin morphisms for lci morphisms and prove an adjunction theorem linking our theory to Rost's. This also extends Schmid's thesis ([27]).

We give a new proof of the fact that Milnor-Witt K-theory has geometric transfers. The proof yields to a simplification of Morel's conjecture about transfers on contracted homotopy sheaves.

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This article is a sequel of [Fel18]. We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus generalising Déglise's thesis [Dé11]. As a corollary, we recover a theorem of Ananyevskiy and Neshitov [AN18] and we prove that Chow-Witt groups in degree zero are birational invariants.