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In this paper we address the task of recognizing assembly actions as a structure (e.g. a piece of furniture or a toy block tower) is built up from a set of primitive objects. Recognizing the full range of assembly actions requires perception at a level of spatial detail that has not been attempted in the action recognition literature to date. We extend the fine-grained activity recognition setting to address the task of assembly action recognition in its full generality by unifying assembly actions and kinematic structures within a single framework. We use this framework to develop a general method for recognizing assembly actions from observation sequences, along with observation features that take advantage of a spatial assembly's special structure. Finally, we evaluate our method empirically on two application-driven data sources: (1) An IKEA furniture-assembly dataset, and (2) A block-building dataset. On the first, our system recognizes assembly actions with an average framewise accuracy of 70% and an average normalized edit distance of 10%. On the second, which requires fine-grained geometric reasoning to distinguish between assemblies, our system attains an average normalized edit distance of 23% -- a relative improvement of 69% over prior work.