  Many interesting examples of operator algebras, both self-adjoint and
non-self-adjoint, can be constructed from directed graphs. In this survey, we
overview the construction of $C^*$-algebras from directed graphs and from two
generalizations of graphs: higher rank graphs and categories of paths. We also
look at free semigroupoid algebras generated from graphs and higher rank
graphs, with an emphasis on the left regular free semigroupoid algebra. We give
examples of specific graphs and the algebras they generate, and we discuss
properties such as semisimplicity and reflexivity. Finally, we propose a new
construction: applying the left regular free semigroupoid construction to
categories of paths.
