  We introduce a scheme-theoretic enrichment of the principal objects of
tropical geometry. Using a category of semiring schemes, we construct tropical
hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} =
(\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution
sets to explicit systems of tropical equations that are uniquely determined by
idempotent module theory. We then define a tropicalization functor that sends
closed subschemes of a toric variety over a ring R with non-archimedean
valuation to closed subschemes of the corresponding tropical toric variety.
Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's
extended tropicalization, and in the case of a projective hypersurface we show
that the scheme structure determines the multiplicities attached to the
top-dimensional cells. By varying the valuation, these tropicalizations form
algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of
valuations on R that we construct. For projective subschemes, the Hilbert
polynomial is preserved by tropicalization, regardless of the valuation. We
conclude with some examples and a discussion of tropical bases in the
scheme-theoretic setting.
