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Neural Radiance Fields (NeRF) are able to reconstruct scenes with unprecedented fidelity, and various recent works have extended NeRF to handle dynamic scenes. A common approach to reconstruct such non-rigid scenes is through the use of a learned deformation field mapping from coordinates in each input image into a canonical template coordinate space. However, these deformation-based approaches struggle to model changes in topology, as topological changes require a discontinuity in the deformation field, but these deformation fields are necessarily continuous. We address this limitation by lifting NeRFs into a higher dimensional space, and by representing the 5D radiance field corresponding to each individual input image as a slice through this "hyper-space". Our method is inspired by level set methods, which model the evolution of surfaces as slices through a higher dimensional surface. We evaluate our method on two tasks: (i) interpolating smoothly between "moments", i.e., configurations of the scene, seen in the input images while maintaining visual plausibility, and (ii) novel-view synthesis at fixed moments. We show that our method, which we dub HyperNeRF, outperforms existing methods on both tasks. Compared to Nerfies, HyperNeRF reduces average error rates by 4.1% for interpolation and 8.6% for novel-view synthesis, as measured by LPIPS. Additional videos, results, and visualizations are available at https://hypernerf.github.io.