  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.
