Let $G$ be a finite abelian group of order $n$ (written additively), and let $S=(a_1, a_2, \cdots, a_k)$ be a sequence of elements in $G$ (repeatition allowed). Let $\sigma (S)$ denote the sum $\sum_{i=1}^ka_i$. We call $S$ a {\it zero-sum sequence} if $\sigma(S)=0$ (the identify element of $G$). Problems in zero-sum are usually determining the minimal length that a sequence should have to ensure it contains a zero-sum subsequence (resp. of restricted size, or containing some special element), or descibing the structure of sequences of long lengths which contain no zero-sum subsequence (resp. of restricted size, or containning some special element). Many problems in zero-sum are connected with factorizations in algebraic number theory, and the others are mostly related to the so called Erd\H{o}s-Ginzburg-Ziv theorem. The most important one among the former problems is the Davenport constant $D(G)$ of $G$, which is the smallest integer $d$ such that every sequence of $d$ elements in $G$ contains a zero-sum subsequence. $D(G)$ was first formulated by H. Davenport in Michigan Number Theory Conference in 1966, and he pointed out that $D(G)$ has some conneting with algebraic number theory in the follwoing way: Let $K$ be a number field, $R$ its algebraic integers ring, and $G$ its class group. Then, $D(G)$ is the maximal number of prime ideals (countng multiplicities) that can occur in a decomposition of an irreducible element of $R$. The Erd\H{o}s-Ginzburg-Ziv theorem states that, every sequence of $2n-1$ elements in $G$ conatins a zero-sum subsequence of length $n$. In this talk, we shall give a survey of results around the Davenport constant and the Erd\H{o}s-Ginzburg-Ziv theorem in recent twenty years, and some open problems are presented as well.