We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. We will then describe some refinements and applications. Refinements include a Riemann-Roch theorem for tropical curves, proved by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for metrized complexes of curves, proved by Amini and the speaker. Applications include a new proof by Cools-Draisma-Payne-Robeva of the Brill-Noether theorem in algebraic geometry, a generalization by Amini and the speaker of the Eisenbud-Harris theory of limit linear series, and a new Chabauty-Coleman style bound for the number of rational points on an algebraic curve over the rationals, proved recently by Katz and Zureick-Brown.