Classical Brill-Noether theory addresses the existence of special divisors on a general curve of genus g.  In particular, Griffiths, Harris, Kleiman and Laksov combined to show that a general curve of genus g has a divisor of degree d and rank r if and only if g is at least (r+1)(g-d+r).  Baker and Norine introduced a theory of ranks of divisors on graphs and metric graphs that is closely linked to ranks of divisors on algebraic curves, through specialization.  Furthermore, Baker conjectured that suitable versions of Brill-Noether theorems should exist in tropical geometry and showed that some Brill-Noether type theorems on algebraic curves could be reproved tropically.  In this talk, I will present recent joint work with Filip Cools, Jan Draisma, and Elina Robeva exploring these topics.