Given an additively written abelian group G and a set X ⊆ G, we let (X) denote the monoid of zero-sum sequences over X and (X) the Davenport constant of (X), namely the supremum of the positive integers n for which there exists a sequence x₁⋯xₙ in (X) such that ${\sum }_{i\in I}{x}_i\ne 0$ for each non-empty proper subset I of 1,...,n. In this paper, we mainly investigate the case when G is a power of ℤ and X is a box (i.e., a product of intervals of G). Some mixed sets (e.g., the product of a group by a box) are studied...