One of the most basic classes of algorithmic problems in combinatorial optimization is the computation of shortest paths. Tropical geometry is a natural language to analyze parametrized versions where some of the arc weights are unknown. I will introduce this point of view, with a link to polyhedral geometry. Moreover, I will present two applications for those graphs, traffic networks and the enumeration of polytropes, building blocks of tropical convexity. This is ongoing work with Michael Joswig.