The Jacobian Pic$^0$(C) of a tropical curve C of genus g is a g-dimensional real torus which acts simply transitively on Pic$^\mathrm{d}$(C) for every d. If C is the geometric realization of a finite graph G, Pic$^0$(C) contains Pic$^0$(G) as a subgroup; the latter is a finite abelian group whose cardinality is the number of spanning trees in G. If G is planar, there is a canonical simply transitive action of Pic$^0$(G) on the set of spanning trees of G; in a certain precise sense this characterizes planar graphs. In order to explain this, I will introduce the important concept of break divisors on graphs and tropical curves, which form a canonical set of effective representatives for Pic$^\mathrm{g}$(G) and Pic$^\mathrm{g}$(C), respectively. Break divisors on graphs are intimately connected with Gioan’s circuit-cocircuit reversal system, which provides a useful combinatorial interpretation of Pic$^{\mathrm{g-1}}$(G). I will describe an interesting family of bijections between circuit-cocircuit reversal classes and spanning trees whose properties are best understood through the geometric theory of graphical zonotopes and their tilings. If time permits, I will explain how such bijections extend naturally from graphs to regular matroids.