Hall's marriage theorem

In 1935, Philip Hall published what is often referred to as ``Hall’s marriage theorem’’ in a short paper (P.~Hall, On Representatives of Subsets, $\mathit{\text{J.}}\mathit{\text{Lond.}}\mathit{\text{Math.}}\mathit{\text{Soc.}}$ (1) $\mathbf{\text{10}}$ (1935), no.1, 26–30.) This paper has been very influential. I state the theorem and outline Hall’s proof, together with some equivalent (or stronger) earlier results, and proceed to discuss some the many directions in combinatorics and beyond which this theorem has influenced.