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Model-based design encourages architects to use customized architectural languages that precisely target their problem domains. These languages, coined domain-specific modeling languages (DSMLs), are essential components of model-driven architecture (MDA), platform-based design, and model-integrated computing (MIC). As a result, tools and methods for DSMLs have blossomed: Key research includes the heterogeneous composition of DSMLs, model transformations between DSMLs, approximations of formal properties within DSMLs, and reuse of DSML semantics. Despite this success, some key foundational issues concerning DSMLs have not been addressed. My presentation will address one of these issues; the structural semantics of DSMLs. The structural semantics of some DSML describes the set of structurally well-formed models that belong to that language. I will describe a formalization of structural semantics that takes into account current metamodeling techniques, provides a formal understanding of metacircularity, and allows us to algorithmically decide if two metamodels are equivalent. I will also describe how this theory has been used to provide a formal underpinning for existing metaprogrammable tools, providing proof-theoretic methods for evaluating DSML properties. ©2007 Microsoft Corporation. All rights reserved.