Let G be a finite abelian group. A zero-sum problem on G asks for
the smallest positive integer k such that for any sequence a_1,...,a_k
of elements of G there exists a subsequence of required length the sum of
whose terms vanishes. In this talk we will give a survey of problems and
results in this field. In particular, we will talk about Olson's theorem
on the Davenport constanst of an abelian p-group and Reiher's celebrated
proof of the Kemnitz conjecture.