I’ll discuss joint work with J.H. Giansiracusa (Swansea) in which we study scheme theory over the tropical semiring \mathbb{T}, using the notion of semiring schemes provided by Toen-Vaquie, Durov, or Lorscheid. We define tropical hypersurfaces in this setting and a tropicalization functor that sends closed subschemes of a toric variety over a field with non-archimedean valuation to closed subschemes of the corresponding toric variety over \mathbb{T}. Upon passing to the set of \mathbb{T}-valued points this yields Payne’s extended tropicalization functor. We prove that the Hilbert function of any projective subscheme is preserved by our tropicalization functor, so the scheme-theoretic foundations developed here reveal a hidden flatness in the degeneration sending a variety to its tropical skeleton.