A group $G$ admits an \textbf{\em $n$-partite digraphical representation} if there exists a regular $n$-partite digraph $Γ$ such that the automorphism group $\mathrm{Aut}(Γ)$ of $Γ$ satisfies the following properties: $\mathrm{Aut}(Γ)$ is isomorphic to $G$, $\mathrm{Aut}(Γ)$ acts semiregularly on the vertices of $Γ$ and the orbits of $\mathrm{Aut}(Γ)$ on the vertex set of $Γ$ form a partition into $n$ parts giving a structure of $n$-partite digraph to $Γ$. In this paper, for every positive integer $n$, we classify the finite groups admitting an $n$-partite digraphical representation.