Theories of object recognition agree that shape is of primordial importance, but there is no consensus about how shape might be represented, and so far attempts to implement a model of shape perception that would work with realistic stimuli have largely failed. Recent studies suggest that state-of-the-art convolutional 'deep' neural networks (DNNs) capture important aspects of human object perception. We hypothesized that these successes might be partially related to a human-like representation of object shape. Here we demonstrate that sensitivity for shape features, characteristic to human and primate vision, emerges in DNNs when trained for generic object recognition from natural photographs. We show that these models explain human shape judgments for several benchmark behavioral and neural stimulus sets on which earlier models mostly failed. In particular, although never explicitly trained for such stimuli, DNNs develop acute sensitivity to minute variations in shape and to non-accidental properties that have long been implicated to form the basis for object recognition. Even more strikingly, when tested with a challenging stimulus set in which shape and category membership are dissociated, the most complex model architectures capture human shape sensitivity as well as some aspects of the category structure that emerges from human judgments. As a whole, these results indicate that convolutional neural networks not only learn physically correct representations of object categories but also develop perceptually accurate representational spaces of shapes. An even more complete model of human object representations might be in sight by training deep architectures for multiple tasks, which is so characteristic in human development.