We propose to represent shapes as the deformation and combination oflearnable elementary 3D structures, which are primitives resulting fromtraining over a collection of shape. We demonstrate that the learned elementary3D structures lead to clear improvements in 3D shape generation and matching.More precisely, we present two complementary approaches for learning elementarystructures: (i) patch deformation learning and (ii) point translation learning.Both approaches can be extended to abstract structures of higher dimensions forimproved results. We evaluate our method on two tasks: reconstructing ShapeNetobjects and estimating dense correspondences between human scans (FAUST interchallenge). We show 16% improvement over surface deformation approaches forshape reconstruction and outperform FAUST inter challenge state of the art by6%.