It is desirable to enable robots capable of automatic assembly. Structural understanding of object parts plays a crucial role in this task yet remains relatively unexplored. In this paper, we focus on the setting of furniture assembly from a complete set of part geometries, which is essentially a 6-DoF part pose estimation problem. We propose a multi-layer transformer-based framework that involves geometric and relational reasoning between parts to update the part poses iteratively. We carefully design a unique instance encoding to solve the ambiguity between geometrically-similar parts so that all parts can be distinguished. In addition to assembling from scratch, we extend our framework to a new task called in-process part assembly. Analogous to furniture maintenance, it requires robots to continue with unfinished products and assemble the remaining parts into appropriate positions. Our method achieves far more than 10% improvements over the current state-of-the-art in multiple metrics on the public PartNet dataset. Extensive experiments and quantitative comparisons demonstrate the effectiveness of the proposed framework.