We propose to represent shapes as the deformation and combination of learnt elementary 3D structures. We demonstrate this decomposition in learnt elementary 3D structures is highly interpretable and leads to clear improvements in 3D shape generation and matching. More precisely, we present two complementary approaches to learn elementary structures in a deep learning framework: (i) continuous surface deformation learning and (ii) 3D structure points learning. Both approaches can be extended to abstract structures of higher dimensions for improved results. We evaluate our method on two very different tasks: ShapeNet objects reconstruction and dense correspondences estimation between human scans. Qualitatively our approach provides interpretable and repeatable results. Quantitatively, we show an important 16% boost for 3D object generation via surface deformation, as well as a clear 6% improvement over state of the art correspondence results on the FAUST inter challenge.