If f : S' ͺ S is a finite locally free morphism of schemes, we construct a symmetric monoidal « norm » functor f (x) : H.(S) ͺ H.(S), where H.(S) is the pointed unstable motivic homotopy category over S. If f is finite étale, we show that it stabilizes to a functor f(x) : SH.(S') ͺ SH(S), where SH(S) is the P¹ -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_∞-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 HZ , the homotopy K -theory spectrum KGL , and the algebraic cobordism spectrum MGL . The normed spectrum structure on HZ is a common refinement of Pulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.