We often build operator algebras from combinatorial objects such as directed graphs and higher-rank analogues. Directed graphs consist of a set of vertices and a set of directed edges pointing from one vertex to another. To build an operator algebra from a directed graph, the vertices become mutually orthogonal subspaces of a fixed infinite-dimensional space, and the edges become operators between the subspaces. The ``graph algebras'' generated by these operators have attracted an enormous amount of interest over the last 15 years because they provide rich classes of tractable examples on which we can test hypotheses and theories. I will give an introduction to the key ideas, and explain the recent role of the operator algebras of higher-rank graphs in the classification of simple algebras.