Let G be a finite abelian group (written additively), and let D (G) denote the Davenport's constant of G, i.e. the smallest integer d such that every sequence of d elements (repetition allowed) in G contains a nonempty zero-sum subsequence. Let S be a sequence of elements in G with |S|≥D (G). We say S is a normal sequence if S contains no zero-sum subsequence of length larger than |S|−D(G)+1. In this paper we obtain some results on the structure of normal sequences for arbitrary G. If G=Cn⊕Cn and nsatisfies some well-investigated property, we determine all normal sequences. Applying these results, we obtain correspondingly some results on the structure of the sequence S in G of length |S|=|G|+D(G)−2 and S contains no zero-sum subsequence of length |G|.