Tropical geometry of moduli spaces of weighted stable curves Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogues of the tautological maps, investigate the dependence of the tropical moduli spaces on the weight data, and consider the example of Losev-Manin spaces.