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A finite group G $G$ admits an oriented regular representation if there exists a Cayley digraph of G $G$ such that it has no digons and its automorphism group is isomorphic to G $G$. Let m $m$ be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented m $m$‐semiregular representations using m $m$‐Cayley digraphs. Given a finite group G $G$, an m $m$‐Cayley digraph of G $G$ is a digraph that has a group of automorphisms isomorphic to G $G$ acting semiregularly on the vertex set with m $m$ orbits. We say that a finite group G $G$ admits an orientedm $m$‐semiregular representation (Om $m$SR for short) if there exists an m $m$‐Cayley digraph Γ ${\rm{\Gamma }}$ of G $G$ such that it has no digons and G $G$ is isomorphic to its automorphism group. Moreover, if Γ ${\rm{\Gamma }}$ is regular, that is, each vertex has the same in‐ and out‐valency, we say Γ ${\rm{\Gamma }}$ is a regular orientedm $m$‐semiregular representation (regular Om $m$SR for short) of G $G$. In this paper, we classify finite groups admitting a regular Om $m$SR or an Om $m$SR for each positive integer m $m$. [ABSTRACT FROM AUTHOR]