The authors study the structure of sequences in \(\mathbb{Z}_n^d\) that avoid zero-sums of length \(n\). Let \(s(\mathbb{Z}_n^d)\) denote the least number \(s\) so that every sequence of \(s\) elements in \(\mathbb{Z}_n^d\) contains \(n\) elements whose sum is \(0 \in \mathbb{Z}_n^d\). If \(d=1\) for example, the Erdős-Ginzburg-Ziv theorem [\textit{P.~Erdős, A. Ginzburg} and \textit{A. Ziv}, ``Theorem in the additive number theory'', Bull. Res. Council Israel 10, 41--43 (1961; Zbl 0063.00009)] states that an arbitrary sequence of \(2n-1\) elements of \(\mathbb{Z}_n\) has a subsequence of \(n\) elements, whose sum is \(0\). Moreover, a sequence of only \(2n-2\) elements without a zero-sum of length \(n\) necessarily consists of \(n-1\) copies of two distinct elements [\textit{T.~Yuster} and \textit{B.~Peterson}, Can. J. Math. 36, No. 3, 529--536 (1984; Zbl 0552.10034)]. The authors study these questions for larger \(d\). It is expected that similar phenomena hold. Very recently Christian Reiher proved Kemnitz conjecture, i.e. that \(s(\mathbb{Z}_n^2)=4n-3\). For odd \(n\), the reviewer [Lower bounds for multidimensional zero sums. Combinatorica 24, No. 3, 351--358 (2004)] showed \(s(\mathbb{Z}_n^d)\geq (1.125)^{\lfloor d/3 \rfloor} 2^d (n-1)+1\), while \textit{N. Alon} and \textit{M. Dubiner} [Combinatorica 15, No. 3, 301--309 (1995; Zbl 0838.11020)] proved that \(s(\mathbb{Z}_n^d) \leq ( c d \log d)^d n\) with an absolute constant \(c\). The authors state a number of conjectures and properties of sequences and show how these are related. In particular: let \(c(n,d)=\frac{s(\mathbb{Z}_n^d)-1}{n-1}\). For those few examples known so far, \(c(n,d)\) is an integer and the extremal cases of sequences of length \(s(\mathbb{Z}_n^d)-1\) without a zero-sum of length \(n\) consist of \(n-1\) copies of \(c(n,d)\) distinct elements. The authors prove this for \(n=2^a\) and arbitrary \(d\), and for \(n=3^a\) and \(d=3\), and with \(c(2^a,d)=2^d, c(3^a,3)=9\). This paper gives a lot of new insight how the extremal cases of this and related problems in \(\mathbb{Z}_n^d\) might look like.