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The light transport (LT) of a scene describes how it appears under different lighting and viewing directions, and complete knowledge of a scene's LT enables the synthesis of novel views under arbitrary lighting. In this paper, we focus on image-based LT acquisition, primarily for human bodies within a light stage setup. We propose a semi-parametric approach to learn a neural representation of LT that is embedded in the space of a texture atlas of known geometric properties, and model all non-diffuse and global LT as residuals added to a physically-accurate diffuse base rendering. In particular, we show how to fuse previously seen observations of illuminants and views to synthesize a new image of the same scene under a desired lighting condition from a chosen viewpoint. This strategy allows the network to learn complex material effects (such as subsurface scattering) and global illumination, while guaranteeing the physical correctness of the diffuse LT (such as hard shadows). With this learned LT, one can relight the scene photorealistically with a directional light or an HDRI map, synthesize novel views with view-dependent effects, or do both simultaneously, all in a unified framework using a set of sparse, previously seen observations. Qualitative and quantitative experiments demonstrate that our neural LT (NLT) outperforms state-of-the-art solutions for relighting and view synthesis, without separate treatment for both problems that prior work requires.