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Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$, and let $\overline{X}$ and $\overline{Y}$ 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 Br $\left(\overline{X}\right)\oplus$ Br($\overline{Y})$ has finite index in the Galois invariant subgroup of Br$(\overline{X}\times \overline{Y})$. This implies that the cokernel of the natural map Br$\left(X\right)\oplus$ Br$\left(Y\right)\to$ 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...