I will survey some old work initiated by Pitts and myself about the class of free semigroup algebras. The prototype is the left regular representation of the free semigroup on $n$ letters acting on full Fock space. We called the weak operator closed algebra it generates the non-commutative analytic Toeplitz algebra because of the striking parallels with the theory of the shift operator. While we and also Popescu found some deep connections, recent work by Jury, Martin, Shamovich and others has revealed even more connections. I can't do justice to all of this material, so I will set the stage by mostly talking about the older work that I was involved in, and provide some hints about why you should investigate the recent work.