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We present TOCH, a method for refining incorrect 3D hand-object interaction sequences using a data prior. Existing hand trackers, especially those that rely on very few cameras, often produce visually unrealistic results with hand-object intersection or missing contacts. Although correcting such errors requires reasoning about temporal aspects of interaction, most previous works focus on static grasps and contacts. The core of our method are TOCH fields, a novel spatio-temporal representation for modeling correspondences between hands and objects during interaction. TOCH fields are a point-wise, object-centric representation, which encode the hand position relative to the object. Leveraging this novel representation, we learn a latent manifold of plausible TOCH fields with a temporal denoising auto-encoder. Experiments demonstrate that TOCH outperforms state-of-the-art 3D hand-object interaction models, which are limited to static grasps and contacts. More importantly, our method produces smooth interactions even before and after contact. Using a single trained TOCH model, we quantitatively and qualitatively demonstrate its usefulness for correcting erroneous sequences from off-the-shelf RGB/RGB-D hand-object reconstruction methods and transferring grasps across objects.