Let X and Y be smooth and projective varieties over a field k finitely generated over \mathbb Q , and let $$ and \bar Y be the varieties over an algebraic closure of k obtained from X and Y , respectively, by extension of the ground field. We show that the Galois invariant subgroup of \mathrm{Br}(\bar X)\oplus \mathrm{Br}(\bar Y) has finite index in the Galois invariant subgroup of \mathrm{Br}(\bar X\times\bar Y) . This implies that the cokernel of the natural map \mathrm{Br} (X)\oplus\mathrm{Br} (Y)\to \mathrm{Br}(X\times Y) is finite when k is a number field. In this case we prove that the Brauer–Manin set of the product of varieties is the product of their Brauer–Manin sets.