Texture cues on 3D objects are key to compelling visual representations, with the possibility to create high visual fidelity with inherent spatial consistency across different views. Since the availability of textured 3D shapes remains very limited, learning a 3D-supervised data-driven method that predicts a texture based on the 3D input is very challenging. We thus propose Texturify, a GAN-based method that leverages a 3D shape dataset of an object class and learns to reproduce the distribution of appearances observed in real images by generating high-quality textures. In particular, our method does not require any 3D color supervision or correspondence between shape geometry and images to learn the texturing of 3D objects. Texturify operates directly on the surface of the 3D objects by introducing face convolutional operators on a hierarchical 4-RoSy parametrization to generate plausible object-specific textures. Employing differentiable rendering and adversarial losses that critique individual views and consistency across views, we effectively learn the high-quality surface texturing distribution from real-world images. Experiments on car and chair shape collections show that our approach outperforms state of the art by an average of 22% in FID score.