Let G be an additive finite abelian group and let S = a1 · . . . · al be a sequence over G. We say that S is a zero-sum sequence if a1 + . . . + al = 0. A typical direct zero-sum problem studies conditions which ensure that a sequence contains a zero-sum subsequence of prescribed length (usual conditions require the length to be |G|, or exp(G) or at most exp(G)). The associated inverse zero-sum problem studies the structure of extremal sequences which have no such zero-sum subsequences. These investigations were initiated by a result of P. Erd˝os, A. Ginzburg and A. Ziv in 1961, who proved that 2n − 1 is the smallest integer l ∈ N such that every sequence S over a cyclic group of order n with length |S| ≥ l has a zero-sum subsequence of length n. A classical invariant is the Davenport constant D(G) which is defined as the smallest integer l ∈ N such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence. Zero-sum problems arise naturally in various branches of combinatorics, graph theory, classical number theory and finite geometry. Moreover, zero-sum theory has been promoted by applications in the theory of non-unique factorizations. We give a survey on the variety of zero-sum problems and present some recent results.