This will be an introduction to the geometry tropical curve in analogy with the classical geometry of Riemann surfaces. Our focus will be on the analogous theories of divisors and linear systems on both sides and, in particular, on the connection between them through the process of tropicalization. The central problem is the so-called realizability problem for tropical divisors.
Roughly speaking, it asks which tropical divisors of a certain rank are tropicalizations of algebraic/geometric divisors of the same rank. Secretly, this is a problem about the compactification of Brill-Noether strata in the moduli space of pointed stable curves.
Most of the content of this talk is well-known in the community. What little is new will be based on joint work with M. Brandt as well as M. Möller and A. Werner.