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Let $V$ be a complex vector space of dimension $n\ge 2$  and $K$ be a subset of $\bigwedge^2V$ of dimension $m$. Denote the Koszul module by $W(V,K)$ and its corresponding resonance variety by $\mathcal R(V,K)$. Papadima and Suciu showed that there exists a uniform bound $q(n,m)$ such that the graded component of the Koszul module $W_q(V,K)=0$ for all $q\ge q(n,m)$ and for all $(V,K)$ satisfying $\mathcal R(V,K)=\{0\}$. In this talk, we will determine this bound $q(n,m)$ precisely, and find an upper bound for the Hilbert series of these Koszul modules. Then we will consider a class of Koszul modules associated to vector bundles.