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We introduce NeuralMLS, a space-based deformation technique, guided by a set of displaced control points. We leverage the power of neural networks to inject the underlying shape geometry into the deformation parameters. The goal of our technique is to enable a realistic and intuitive shape deformation. Our method is built upon moving least-squares (MLS), since it minimizes a weighted sum of the given control point displacements. Traditionally, the influence of each control point on every point in space (i.e., the weighting function) is defined using inverse distance heuristics. In this work, we opt to learn the weighting function, by training a neural network on the control points from a single input shape, and exploit the innate smoothness of neural networks. Our geometry-aware control point deformation is agnostic to the surface representation and quality; it can be applied to point clouds or meshes, including non-manifold and disconnected surface soups. We show that our technique facilitates intuitive piecewise smooth deformations, which are well suited for manufactured objects. We show the advantages of our approach compared to existing surface and space-based deformation techniques, both quantitatively and qualitatively.