Tropical geometry studies a piecewise linear combinatorial shadow of degenerations and compactifications of algebraic varieties. A typical phenomenon is that many of the usual algebro-geometric objects have a tropical analogue that is intimately tied to its classical counterpart. An example is the theory of divisors and line bundles on algebraic curves, whose tropical counterparts have been crucial in numerous surprising applications to classical Brill—Noether theory and the birational geometry of moduli spaces. One classical object that has resisted the effort of tropical geometers so far is the geometry of vector bundles beyond rank one. In this talk, I will outline an elementary approach to tropical vector bundles that builds on earlier work of Allermann. Although limited in scope, this theory leads to a satisfying tropical story for semistable vector bundles on elliptic curves and, more generally, semihomogeneous vector bundles on abelian varieties. The engines in the background that make these cases accessible to our methods are Atiyah's classification of vector bundles on elliptic curves, Fourier-Mukai transforms on abelian varieties, and the interactions with non-Archimedean uniformization. This talk is based on joint work with Andreas Gross and Dmitry Zakharov (and, in parts, Arne Kuhrs) as well as with Andreas Gross, Inder Kaur, and Annette Werner.