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Tropical geometry, in which algebraic objects are replaced with piecewise-linear ones, has become an important tool for studying moduli spaces. For example, moduli spaces of “tropical curves” (which are simply metric graphs) turn out to reflect some of the complicated geometry of moduli spaces of algebraic curves. Recent developments in involving "tropical schemes” point toward to a parallel strategy for studying Hilbert schemes. Tropical schemes are closely related to the notion of a matroid, and many basic properties and examples are not yet well-understood. I will discuss the interplay between Hilbert schemes and tropical schemes in the 2-variable case. This includes joint work with Diane Maclagan.