We introduce the scale axis transform, a new skeletal shape representation for bounded open sets O ⊂ &Rdbl;^d. The scale axis transform induces a family of skeletons that captures the important features of a shape in a scale-adaptive way and yields a hierarchy of successively simplified skeletons. Its definition is based on the medial axis transform and the simplification of the shape under multiplicative scaling: the s-scaled shape Os is the union of the medial balls of O with radii scaled by a factor of s. The s-scale axis transform of O is the medial axis transform of Os, with radii scaled back by a factor of 1/s. We prove topological properties of the scale axis transform and we describe the evolution s &rarr Os by defining the multiplicative distance function to the shape and studying properties of the corresponding steepest ascent flow. All our theoretical results hold for any dimension. In addition, using a discrete approximation, we present several examples of two-dimensional scale axis transforms that illustrate the practical relevance of our new framework. © 2009 ACM.