We present a new local descriptor for 3D shapes, directly applicable to a wide range of shape analysis problems such as point correspondences, semantic segmentation, affordance prediction, and shape-to-scan matching. The descriptor is produced by a convolutional network that is trained to embed geometrically and semantically similar points close to one another in descriptor space. The network processes surface neighborhoods around points on a shape that are captured at multiple scales by a succession of progressively zoomed out views, taken from carefully selected camera positions. We leverage two extremely large sources of data to train our network. First, since our network processes rendered views in the form of 2D images, we repurpose architectures pre-trained on massive image datasets. Second, we automatically generate a synthetic dense point correspondence dataset by non-rigid alignment of corresponding shape parts in a large collection of segmented 3D models. As a result of these design choices, our network effectively encodes multi-scale local context and fine-grained surface detail. Our network can be trained to produce either category-specific descriptors or more generic descriptors by learning from multiple shape categories. Once trained, at test time, the network extracts local descriptors for shapes without requiring any part segmentation as input. Our method can produce effective local descriptors even for shapes whose category is unknown or different from the ones used while training. We demonstrate through several experiments that our learned local descriptors are more discriminative compared to state of the art alternatives, and are effective in a variety of shape analysis applications.