  Given finitely generated modules $M$ and $N$ over a local ring $R$, the
tensor product $M\otimes_RN$ typically has nonzero torsion. Indeed, the
assumption that the tensor product is torsion-free influences the structure and
vanishing of the modules $\Tor^R_i(M,N)$ for all $i\geq 1$. In turn, the
vanishing of $\Tor^R_i(M,N)$ imposes restrictions on the depth properties of
the modules $M$ and $N$. These connections made their first appearance in
Auslander's 1961 paper "Modules over unramified regular local rings". We will
survey the literature on these topics, with emphasis on progress during the
past twenty years.
