Title: Neural Photometric Stereo for Shape and Material Estimation. Abstract: This paper addresses a challenging Photometric-Stereo problem where the object to be reconstructed has unknown, non-Lambertian, and possibly spatially-varying surface materials. This problem becomes even more challenging when the shape of the object is highly complex so that shadows cast on the surface are inevitable.  To overcome these challenges, we propose a simple coordinate-based deep MLP (multilayer perceptron) neural network to parameterize both the unknown 3D shape and the unknown spatially-varying reflectance at every image pixel. This network is able to leverage the observed specularities and shadows on the surface, and recover both surface shape, normal and generic non-Lambertian reflectance via an inverse differentiable rendering process.  We explicitly predict cast shadows, mitigating possible artifacts on these shadowing regions, leading to higher estimation accuracy. Our framework is entirely self-supervised, in the sense that it requires neither ground truth shape nor known svBRDF. Tests on real-world images demonstrate that our method achieves state-of-the-art accuracy in both shape recovery and material estimation. Thanks to the small size of the MLP-net, our method is also an order of magnitude faster than previous competing deep-learning based photometric stereo methods.