  We develop the theory of Milnor-Witt motives and motivic cohomology. Compared
to Voevodsky's theory of motives and his motivic cohomology, the first
difference appears in our definition of Milnor-Witt finite correspondences,
where our cycles come equipped with quadratic forms. This yields a weaker
notion of transfers and a derived category of motives that is closer to the
stable homotopy theory of schemes. We prove a cancellation theorem when
tensoring with the Tate object, we compare the diagonal part of our Milnor-Witt
motivic cohomology to Minor-Witt K-theory and we provide spectra representing
various versions of motivic cohomology in the $\mathbb{A}^1$-derived category
or the stable homotopy category of schemes.
