Let XX and YY be smooth and projective varieties over a field kk finitely generated over \Q\Q, and let \ovX\ovX and \ovY\ovY be the varieties over an algebraic closure of kk obtained from XX and YY, respectively, by extension of the ground field. We show that the Galois invariant subgroup of \Br(\ovX)⊕\Br(\ovY)\Br(\ovX)⊕\Br(\ovY) has finite index in the Galois invariant subgroup of \Br(\ovX×\ovY)\Br(\ovX×\ovY). This implies that the cokernel of the natural map \Br(X)⊕\Br(Y)→\Br(X×Y)\Br(X)⊕\Br(Y)→\Br(X×Y) is finite when kk 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.