  If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a
symmetric monoidal "norm" functor $f_\otimes: \mathcal H_*(S') \to\mathcal
H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable motivic homotopy
category over $S$. If $f$ is finite \'etale, we show that it stabilizes to a
functor $f_\otimes: \mathcal{SH}(S') \to \mathcal{SH}(S)$, where
$\mathcal{SH}(S)$ is the $\mathbb P^1$-stable motivic homotopy category over
$S$. Using these norm functors, we 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: we
investigate the interaction of norms with Grothendieck's Galois theory, with
Betti realization, and with Voevodsky's slice filtration; we prove that the
norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt
rings; and we 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.
