In this talk, I would like to talk about summarize some recent work on how we can decompose geometry into building blocks and reassemble these to create shape variations. The first step is to look at partial correspondences within shapes, where parts of an object are mapped back to itself under a fixed group of transformations. From this matching information, we can compute canonical building blocks that characterize the redundancy in the shape. Further, we can infer tiling grammars that place a constraint on how the pieces found can be assembled to new compound objects if we demand certain similarity between the example geometry and newly created shapes. From this basic framework, a number of applications can be derived, such as fitting of context-free shape grammars to general geometry, or interactive resizing and sculpting. The talk is intended to put some of the work that I have been involved in in the past few years into a common conceptual framework, and show how several different results at the core rely on a few basic principles.