In this paper, we present a method that estimates reflectance and illumination information from a single image depicting a single-material specular object from a given class under natural illumination. We follow a data-driven, learning-based approach trained on a very large dataset, but in contrast to earlier work we do not assume one or more components (shape, reflectance, or illumination) to be known. We propose a two-step approach, where we first estimate the object's reflectance map, and then further decompose it into reflectance and illumination. For the first step, we introduce a Convolutional Neural Network (CNN) that directly predicts a reflectance map from the input image itself, as well as an indirect scheme that uses additional supervision, first estimating surface orientation and afterwards inferring the reflectance map using a learning-based sparse data interpolation technique. For the second step, we suggest a CNN architecture to reconstruct both Phong reflectance parameters and high-resolution spherical illumination maps from the reflectance map. We also propose new datasets to train these CNNs. We demonstrate the effectiveness of our approach for both steps by extensive quantitative and qualitative evaluation in both synthetic and real data as well as through numerous applications, that show improvements over the state-of-the-art.