Tropical geometry provides an array of combinatorial techniques for studying compactifications and degenerations of fundamental objects in algebraic geometry. The piecewise linear objects appearing in tropical geometry are shadows (or skeletons) of nonarchimedean analytic spaces, in the sense of Berkovich, and often capture enough essential information about those spaces to resolve interesting questions about classical algebraic varieties. I will give an overview of tropical geometry as it relates to the study of algebraic curves, touching on applications to moduli spaces of algebraic curves, and refined (or quantum) curve counting as time permits.