Summary: Let \(R\) be a commutative ring and \(M\) an \(R\)-module. In this article, we introduce a new generalization of the annihilating-ideal graph of commutative rings to modules. The annihilating submodule graph of \(M\), denoted by \(\mathbb{G}(M)\), is an undirected graph with vertex set \(\mathbb{A}^*(M)\) and two distinct elements \(N\) and \(K\) of \(\mathbb{A}^*(M)\) are adjacent if \(N*K=0\). In this paper we show that \(\mathbb{G}(M)\) is a connected graph, \(\mathrm{diam}(\mathbb{G}(M))\leq 3\), and \(\mathrm{gr}(\mathbb{G}(M)) \leq 4\) if \(\mathbb{G}(M)\) contains a cycle. Moreover, \(\mathbb{G}(M)\) is an empty graph if and only if \(\mathrm{ann}(M)\) is a prime ideal of \(R\) and \(\mathbb{A}^*(M)\neq \mathbb{S} (M)\setminus \{0\}\) if and only if \(M\) is a uniform \(R\)-module, \(\mathrm{ann}(M)\) is a semi-prime ideal of \(R\) and \(\mathbb{A}^*(M)\neq \mathbb{S} (M) \setminus \{0\}\). Furthermore, \(R\) is a field if and only if \(\mathbb{G}(M)\) is a complete graph, for every \(M\in R-\mathrm{Mod}\). If \(R\) is a domain, for every divisible module \(M\in R-\mathrm{Mod}, \mathbb{G}(M)\) is a complete graph with \(\mathbb{A}^*(M)=\mathbb{S}(M) \setminus \{0\}\). Among other things, the properties of a reduced \(R\)-module \(M\) are investigated when \(\mathbb{G}(M)\) is a bipartite graph.