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There are many conjectures from the representation theory of finite-dimensional algebras that have been transplanted to commutative algebra, and this process has enriched both fields significantly. An example is the celebrated Auslander-Reiten Conjecture, which states that a finitely generated module M over a finite-dimensional algebra A must be projective if $Ext^i_A(M, M) = Ext^i_A(M, A) = 0$ for all $i > 0$. This long-standing conjecture is closely related to other important conjectures such as the Finitistic Dimension Conjecture and Tachikawa Conjecture from representation theory. The Auslander-Reiten Conjecture originates in representation theory of algebras, but it has recently received considerable attention in commutative algebra. In 1994 Huneke and Wiegand proposed a conjecture on tensor products of torsion-free modules over one-dimensional commutative Noetherian integral domains. This conjecture, which is still open, implies the Auslander-Reiten Conjecture for a large class of commutative rings. In this talk I will discuss the connection between the Huneke-Wiegand Conjecture and the Auslander-Reiten Conjecture, and survey the literature on these topics with emphasis on recent progress. Part of the talk is based on an ongoing joint work with Shiro Goto, Ryo Takahashi and Naoki Taniguchi.