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The cohomology groups of complex algebraic varieties come equipped with a powerful but intrinsically analytic invariant called a Hodge structure. Hodge structures of certain very special algebraic varieties are nonetheless parametrized by algebraic varieties, and while this leads to many important applications in algebraic and arithmetic geometry it fails badly in general. Joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman remedies this failure by showing that parameter spaces of Hodge structures always admit "tame" analytic structures in a sense made precise using ideas from model theory. A salient feature of the resulting tame analytic geometry is that it allows for the local flexibility of the full analytic category while preserving the global behavior of the algebraic category.