We study Kummer varieties attached to 2-coverings of abelian varieties of
arbitrary dimension. Over a number field we show that the subgroup of odd
order elements of the Brauer group does not obstruct the Hasse principle.
Sufficient conditions for the triviality of the Brauer group are given, which
allow us to give an example of a Kummer K3 surface of geometric Picard
rank 17 over the rationals with trivial Brauer group. We establish the non-
emptyness of the Brauer–Manin set of everywhere locally soluble Kummer
varieties attached to 2-coverings of products of hyperelliptic Jacobians with
large Galois action on 2-torsion.