Summary: Let \(R\) be a commutative ring with identity and \(M\) an \(R\)-module. In this paper, we associate a graph to \(M\), say \(\Gamma (_{R}M)\), such that when \(M=R\), \(\Gamma (_{R}M)\) coincide with the zero-divisor graph of \(R\). Many well-known results by \textit{D. F. Anderson} and \textit{P. S. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)] have been generalized for \(\Gamma (_{R}M)\). We Will show that \(\Gamma (_{R}M)\) is connected with \(\mathrm{diam}(\Gamma (_{R}M))\leq 3\) and if \(\Gamma (_{R}M)\) contains a cycle, then \(\mathrm{gr}(\Gamma(_{R}M))\leq 4\). We will also show that \(\Gamma(_{R}M)=\emptyset\) if and only if \(M\) is a prime module. Among other results, it is shown that for a reduced module \(M\) satisfying DCC on cyclic submodules, \(\mathrm{gr}(\Gamma(_{R}M))=\infty\) if and only if \(\Gamma(_{R}M)\) is a star graph. Finally, we study the zero-divisor graph of free \(R\)-modules.