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The torsion functors $Tor_i^R(-,-,)$, originally invented by topologists, have become an important tool and a subject of research in commutative algebra. Given finitely generated nonzero modules $M$ and $N$ over a commutative Noetherian local ring $R$, the tensor product $M\otimes_ R N$ of $M$ and $N$ typically has nonzero torsion. For example, if $R$ is the ring of formal power series $\mathbb{C}\mathbb{C}[[X,Y]]$ in indeterminates $X$ and $Y$, and I is the ideal of $R$ generated by $X$ and $Y$, then $I\otimes_R I$ has nonzero torsion. The assumption that the tensor product $M\otimes_RN$ is torsion-free influences the structure and vanishing of the modules $Tor_i^R(M,N)$. In turn, the vanishing of $Tor_i^R(M,N)$ imposes certain restrictions on the properties of $M$ and $N$. These connections made their first appearance in Auslander’s 1961 seminal paper ''Modules over unramified regular local rings''. In this talk I will survey the literature on these topics with emphasis on recent progress. I will also state and discuss several open questions in this direction. The talk is aimed at graduate students.