Abstract: We pose the decompose-and-pack or DAP problem, which tightly combines shape decomposition and packing. While in general, DAP seeks to decompose an input shape into a small number of parts which can be efficiently packed, our focus is geared towards 3D printing. The goal is to optimally decompose-and-pack a 3D object into a printing volume to minimize support material, build time, and assembly cost. We present Dapper, a global optimization algorithm for the DAP problem which can be applied to both powder- and FDM-based 3D printing. The solution search is top-down and iterative. Starting with a coarse decomposition of the input shape into few initial parts, we progressively pack a pile in the printing volume, by iteratively docking parts, possibly while introducing cuts, onto the pile. Exploration of the search space is via a prioritized and bounded beam search, with breadth and depth pruning guided by local and global DAP objectives. A key feature of Dapper is that it works with pyramidal primitives, which are packing- and printing-friendly. Pyramidal shapes are also more general than boxes to reduce part counts, while still maintaining a suitable level of simplicity to facilitate DAP optimization. We demonstrate printing efficiency gains achieved by Dapper, compare to state-of-the-art alternatives, and show how fabrication criteria such as cut area and part size can be easily incorporated into our solution framework to produce more physically plausible fabrications.