Complex vector drawings serve as convenient and expressive visual representations, but they remain difficult to edit or manipulate. For clean-line vector drawings of smooth organic shapes, we describe a method to automatically extract a layered structure for the drawn object from the current or nearby viewpoints. The layers correspond to salient regions of the drawing, which are often naturally associated to 'parts' of the underlying shape. We present a method that automatically extracts salient structure, organized as parts with relative depth orderings, from clean-line vector drawings of smooth organic shapes. Our method handles drawings that contain complex internal contours with T-junctions indicative of occlusions, as well as internal curves that may either be expressive strokes or substructures. To extract the structure, we introduce a new part-aware metric for complex 2D drawings, the radial variation metric, which is used to identify salient parts. These parts are then considered in a priority-ordered fashion, which enables us to identify and recursively process new shape parts while keeping track of their relative depth ordering. The output is represented in terms of scalable vector graphics layers, thereby enabling meaningful sketch editing and manipulation. We evaluate the method on multiple input drawings and show that the structure we compute is convenient for subsequent posing and animation from nearby viewpoints. Lastly, to further demonstrate the usefulness of our system, we develop two direct applications, namely the creation of cardboard articulated puppets and part-based 3D modeling from a single sketch.