Representing a 3D shape with a set of primitives can aid perception of structure, improve robotic object manipulation, and enable editing, stylization, and compression of 3D shapes. Existing methods either use simple parametric primitives or learn a generative shape space of parts. Both have limitations: parametric primitives lead to coarse approximations, while learned parts offer too little control over the decomposition. We instead propose to decompose shapes using a library of 3D parts provided by the user, giving full control over the choice of parts. The library can contain parts with high-quality geometry that are suitable for a given category, resulting in meaningful decom- positions with clean geometry. The type of decomposition can also be controlled through the choice of parts in the library. Our method works via a unsupervised approach that iteratively retrieves parts from the library and refines their placements. We show that this approach gives higher reconstruction accuracy and more desirable decompositions than existing approaches. Additionally, we show how the decom- position can be controlled through the part library by using different part libraries to reconstruct the same shapes.