Let \(M\) be a module over a commutative ring \(R\) with identity. A proper submodule \(P\) with \(p=P:M\) is called a \(p\)-prime submodule if from \(r e \in P\) follows either \(e \in P\) or \(r \in p\), for \(r \in R\), \(e \in M\). Let \(X= \text{Spec}(M)\) be the set of all prime submodules of \(M\), which is called the prime spectrum of \(M\). In the paper is introduced as analogue of the topology on \(\text{Spec}(R)\) a topology (the so-called Zariski topology) on \(X\), in which closed sets are varieties \(V(N)=\{P \in X\mid N:M \subset P:M\}\) of all submodules \(N\) of \(M\). It is shown that \(X\) is a \(T_0\)-space iff \(\psi\) is injective iff \(X\) has at most one \(p\)-prime submodule for every \(p \in \text{Spec}(R)\), where \(\psi\) is the natural map from \(\text{Spec}(M)\) to \(\text{Spec}(R)/\text{Ann}(M)\) defined by \(\psi(P)=(P:M) + \text{Ann}(M)\), \(P \in \text{Spec}(M)\). It is proved that if \(\psi\) is surjective, then \(X\) is an almost spectral space. As an application the author shows that if \(M\) is a finitely generated non-zero \(R\)-module, then \(X\) is a spectral space iff \(M\) is a multiplication module iff \(X\) is homeomorphic to \(\text{Spec}(R/\text{Ann}(M))\) iff \(\psi\) is injective.