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 $Bbb G(M)$, is an undirected graph with vertex set $Bbb A^*(M)$ and two distinct elements $N$ and $K$ of $Bbb A^*(M)$ are adjacent if $N*K=0$. In this paper we show that $Bbb G(M)$ is a connected graph, ${rm diam}(Bbb G(M))leq 3$, and ${rm gr}(Bbb G(M))leq 4$ if $Bbb G(M)$ contains a cycle. Moreover, $Bbb G(M)$ is an empty graph if and only if ${rm ann}(M)$ is a prime ideal of $R$ and $Bbb A^*(M)neq Bbb S(M)setminus {0}$ if and only if $M$ is a uniform $R$-module, ${rm ann}(M)$ is a semi-prime ideal of $R$ and $Bbb A^*(M)neq Bbb S(M)setminus {0}$. Furthermore, $R$ is a field if and only if $Bbb G(M)$ is a complete graph, for every $Min R-{rm Mod}$. If $R$ is a domain, for every divisible module $Min R-{rm Mod}$, $Bbb G(M)$ is a complete graph with $Bbb A^*(M)=Bbb S(M)setminus {0}$. Among other things, the properties of a reduced $R$-module $M$ are investigated when $Bbb G(M)$ is a bipartite graph.