The paper proves a Freiman–Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving a conjecture of Marton. Specifically, for an abelian group \( G \) of torsion \( m \) and a non-empty subset \( A \subseteq G \) with \( |A + A| \leq K|A| \), \( A \) can be covered by at most \( (2K)^{O(m^3)} \) translates of a subgroup \( H \leq G \) of cardinality at most \( |A| \). The argument is a variant of that used in the case \( G = \mathbf{F}_2^n \) in a recent paper by the authors. The main result is stated as Theorem 1.1, and the proof involves the use of multidistance and conditional Ruzsa distance, with key technical results including Proposition 2.3 and Proposition 2.4. The paper also discusses the reduction of the main theorem to these propositions and provides detailed proofs of the necessary lemmas and propositions.The paper proves a Freiman–Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving a conjecture of Marton. Specifically, for an abelian group \( G \) of torsion \( m \) and a non-empty subset \( A \subseteq G \) with \( |A + A| \leq K|A| \), \( A \) can be covered by at most \( (2K)^{O(m^3)} \) translates of a subgroup \( H \leq G \) of cardinality at most \( |A| \). The argument is a variant of that used in the case \( G = \mathbf{F}_2^n \) in a recent paper by the authors. The main result is stated as Theorem 1.1, and the proof involves the use of multidistance and conditional Ruzsa distance, with key technical results including Proposition 2.3 and Proposition 2.4. The paper also discusses the reduction of the main theorem to these propositions and provides detailed proofs of the necessary lemmas and propositions.