a tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. we define the complete linear system $|d|$ of a divisor $d$ on a tropical curve analogously to the classical counterpart. due to work of baker and norine, there is a rank function $r(d)$ on such linear systems, as well a canonical divisor $k$. completely analogous to the classical case, this rank function satisfies riemann-roch and analogues of riemann-hurwitz. after an introduction to these tropical analogues, this talk will describe joint work with josephine yu and christian haase investigating the structure of $|d|$ as a cell complex. we show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. using a finite set of generators, $|d|$ defines a map from the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to $\mathrm{deg}(d)$. the tropical convex hull of the image realizes the linear system $|d|$ as an embedded polyhedral complex.