Directed graphs, variations on directed graphs, and their $C^*$-algebras

We describe some current developments in the use of combinatorial objects to define $C^*$-algebras from a historical point of view. Beginning with the origins of graph $C^*$-algebras, we motivate the definitions for submonoids of groups, and for left cancellative small categories generally, giving the precise construction for the finitely aligned case. We close with a family of interesting examples based on this construction.