In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category C by a set S of morphisms in the heart Hw of a weight structure w on it one obtains a triangulated category endowed with a weight structure w′. The heart of w′ is a certain version of the Karoubi envelope of the non-commutative localization Hw[S^-1]_add (of Hw by S). The functor Hw → Hw[S^-1]_add is the natural categorical version of Cohn's localization of a ring, i.e., it is universal among additive functors that make all elements of S invertible. For any additive category A, taking C = K^b(A) we obtain a very ecient tool for computing A[S^-1]_add; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that A[S^-1]_add coincides with the abstract localization A[S^-1] (as constructed by Gabriel and Zisman) if S contains all identity morphisms of A and is closed with respect to direct sums. We apply our results to certain categories of birational motives DM^o _gm(U) (generalizing those defined by Kahn and Sujatha). We define DM^o _gm(U) for an arbitrary U as a certain localization of K^b(Cor (U)) and obtain a weight structure for it. When U is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general U the result is completely new. The existence of the corresponding adjacent t-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over U.