Abstract: We present the Locally Adaptive Morphable Model (LAMM), a highly flexible Auto-Encoder (AE) framework for learning to generate and manipulate 3D meshes. We train our architecture following a simple self-supervised training scheme in which input displacements over a set of sparse control vertices are used to overwrite the encoded geometry in order to transform one training sample into another. During inference, our model produces a dense output that adheres locally to the specified sparse geometry while maintaining the overall appearance of the encoded object. This approach results in state-of-the-art performance in both disentangling manipulated geometry and 3D mesh reconstruction. To the best of our knowledge LAMM is the first end-to-end framework that enables direct local control of 3D vertex geometry in a single forward pass. A very efficient computational graph allows our network to train with only a fraction of the memory required by previous methods and run faster during inference, generating 12k vertex meshes at $>$60fps on a single CPU thread. We further leverage local geometry control as a primitive for higher level editing operations and present a set of derivative capabilities such as swapping and sampling object parts. Code and pretrained models can be found at https://github.com/michaeltrs/LAMM.