The moduli spaces of stable maps to toric varieties occur naturally in the context of enumerative geometry. While they have several excellent properties, they are nonetheless quite singular, reducible, and even non-equidimensional. In genus 0, the situation becomes markedly improved by adding logarithmic structure to the moduli problem. This yields irreducible toroidal compactifications of the space of maps. In turn, tropical geometry gives strong control over the global geometry of this space. For elliptic maps, logarithmic structures alone do not suffice to desingularize these moduli space. However, in conjunction with insights of Abramovich and Wise, the polyhedral geometry of elliptic tropical curves can be used to construct irreducible toroidal compactifications of the moduli space of elliptic curves in toric varieties, generalizing work of Vakil and Zinger. Time permitting, enumerative applications will be discussed.