The Jacobian Pic0(C) of a tropical curve C of genus g is a g-dimensional real torus which acts simply transitively on Pic\mathrmd(C) for every d. If C is the geometric realization of a finite graph G, Pic0(C) contains Pic0(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 Pic0(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 setof effective representatives for Pic\mathrmg(G) and Pic\mathrmg(C), respectively. Break divisors on graphs are intimately connected with Gioan'scircuit-cocircuit reversal system, which provides a useful combinatorial interpretation of Pic\mathrmg-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 explainhow such bijections extend naturally from graphs to regular matroids.