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Neural representations are popular for representing shapes, as they can be learned form sensor data and used for data cleanup, model completion, shape editing, and shape synthesis. Current neural representations can be categorized as either overfitting to a single object instance, or representing a collection of objects. However, neither allows accurate editing of neural scene representations: on the one hand, methods that overfit objects achieve highly accurate reconstructions, but do not generalize to unseen object configurations and thus cannot support editing; on the other hand, methods that represent a family of objects with variations do generalize but produce only approximate reconstructions. We propose NEUFORM to combine the advantages of both overfitted and generalizable representations by adaptively using the one most appropriate for each shape region: the overfitted representation where reliable data is available, and the generalizable representation everywhere else. We achieve this with a carefully designed architecture and an approach that blends the network weights of the two representations, avoiding seams and other artifacts. We demonstrate edits that successfully reconfigure parts of human-designed shapes, such as chairs, tables, and lamps, while preserving semantic integrity and the accuracy of an overfitted shape representation. We compare with two state-of-the-art competitors and demonstrate clear improvements in terms of plausibility and fidelity of the resultant edits.