If \(f : S\prime \rightarrow S\) is a finite locally free morphism of schemes, the authors construct a symmetric monoidal “norm” functor \(f_{\otimes}:\mathcal{H}_{\bullet}(S\prime)\rightarrow \mathcal{H}_{\bullet}(S)\), where \(\mathcal{H}_{\bullet}(S)\) is the pointed unstable motivic homotopy category over \(S\). If \(f\) is finite étale, the authors show that it stabilizes to a functor \(f_{\otimes}:\mathcal{SH}(S\prime) \rightarrow\mathcal{SH}(S)\), where \(\mathcal{SH}(S)\) is the \(\mathbb{P}^{1}\)-stable motivic homotopy category over \(S\).

Using these norm functors, the authors define the notion of a normed motivic spectrum, which is an enhancement of a motivic \(E_{\infty}\)-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: the authors investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; they prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and they construct normed spectrum structures on the motivic cohomology spectrum\(H\mathbb{Z}\), the homotopy \(K\)-theory spectrum \(KGL\), and the algebraic cobordism spectrum \(MGL\). The normed spectrum structure on \(H\mathbb{Z}\) is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.