We show that the non-Archimedean skeleton of the
d
d
-th symmetric power of a smooth projective algebraic curve
X
X
is naturally isomorphic to the
d
d
-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of
X
X
. The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.