Shape understanding - looking at a shape and intuitively understanding which parts are, e.g., body, arms, legs, toes, and ears - is almost effortless for humans. Training a computer to understand shapes in a similar way, however, presents substantial challenges. This talk will describe a useful mathematical shape model, the Blum medial axis (BMA), and methodologies based on the BMA for automatically decomposing a shape into a hierarchy of parts and determining the similarity between those parts. In 2D, we compare our automated results to human perception data gathered from a massive user study, and also provide some useful applications. Unfortunately, the BMA is notoriously sensitive to noise, which is unavoidable in applications. To address this, we propose geometrically coherent approaches to denoising that provide approximation guarantees for the shape boundary. Finally, because the BMA also provides an interesting example of a Whitney stratified set, we will explore some of the resulting elegant mathematical constructions.