  The Weil restriction functor, introduced in the late fifties, was recently
extended by Karpenko to the category of Chow motives with integral
coefficients. In this article we introduce the noncommutative (=NC) analogue of
the Weil restriction functor, where schemes are replaced by dg algebras, and
extend it to Kontsevich's categories of NC Chow motives and NC numerical
motives. Instead of integer coefficients, we work more generally with
coefficients in a binomial ring. Along the way, we extend Karpenko's functor to
the classical category of numerical motives, and compare this extension with
its NC analogue. As an application, we compute the (NC) Chow motive of the Weil
restriction of every smooth projective scheme whose category of perfect
complexes admits a full exceptional collection. Finally, in the case of central
simple algebras, we describe explicitly the NC analogue of the Weil restriction
functor using solely the degree of the field extension. This leads to a
"categorification" of the classical corestriction homomorphism between Brauer
groups.
