I will explain the construction of a continuous and functorial tropicalization map on the moduli space of logarithmic stable maps to a logarithmically smooth scheme. The basic fuel for this construction is the perfect obstruction theory that gives rise to the virtual fundamental class in logarithmic Gromov-Witten theory. Specializing to the case of toric varieties, this framework gives simple conceptual proofs of several fundamental correspondence principles in tropical geometry. I will explain a tropical version of Vakil's Murphy's Law, which states that the question of which tropical curves are realizable by stable maps to toric varieties is, in general, arbitrarily difficult to answer.