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We introduce NIFT, Neural Interaction Field and Template, a descriptive and robust interaction representation of object manipulations to facilitate imitation learning. Given a few object manipulation demos, NIFT guides the generation of the interaction imitation for a new object instance by matching the Neural Interaction Template (NIT) extracted from the demos in the target Neural Interaction Field (NIF) defined for the new object. Specifically, the NIF is a neural field that encodes the relationship between each spatial point and a given object, where the relative position is defined by a spherical distance function rather than occupancies or signed distances, which are commonly adopted by conventional neural fields but less informative. For a given demo interaction, the corresponding NIT is defined by a set of spatial points sampled in the demo NIF with associated neural features. To better capture the interaction, the points are sampled on the Interaction Bisector Surface (IBS), which consists of points that are equidistant to the two interacting objects and has been used extensively for interaction representation. With both point selection and pointwise features defined for better interaction encoding, NIT effectively guides the feature matching in the NIFs of the new object instances such that the relative poses are optimized to realize the manipulation while imitating the demo interactions. Experiments show that our NIFT solution outperforms state-of-the-art imitation learning methods for object manipulation and generalizes better to objects from new categories.