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submitted for the degree of Doctor of Philosophy I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine mani-folds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space X underlying a non-Archimedean analytic space may locally be recovered from the sheaf |OX | of pointwise valuations of its analytic functions; in other words, (X, |OX |) is a skeleton