Patch-based appearance models are used in a wide range of computer vision applications. To learn such models it has previously been necessary to specify a suitable set of patch sizes and shapes by hand. In the jigsaw model presented here, the shape, size and appearance of patches are learned automatically from the repeated structures in a set of training images. By learning such irregularly shaped ‘jigsaw pieces’, we are able to discover both the shape and the appearance of object parts without supervision. When applied to face images, for example, the learned jigsaw pieces are surprisingly strongly associated with face parts of different shapes and scales such as eyes, noses, eyebrows and cheeks, to name a few. We conclude that learning the shape of the patch not only improves the accuracy of appearance-based part detection but also allows for shape-based part detection. This enables parts of similar appearance but different shapes to be distinguished; for example, while foreheads and cheeks are both skin colored, they have markedly different shapes.