Optical imaging techniques are ubiquitous for the resolution of non-uniformities in gas flows. Planar imaging techniques such as laser-induced fluorescence are well established and applied extensively in turbulent reactive flows, offering both high temporal and spatial resolutions. However, planar imaging suffers from a critical disadvantage, the requirement for spatially continuous optical access over large solid angles in both the excitation and detection paths and this precludes their application in many practical situations, for example those encountered in engine testing. Tomographic absorption spectroscopy, TAS, on the other hand, shares many of the advantages of planar imaging techniques but reduces the demands for optical access, because high quality data can be obtained with sparsely sampled volumes. The technique has unrivalled potential for imaging in harsh environments, for example for in-cylinder/in-chamber engine measurements. TAS is beginning to mature as a technique for the simultaneous imaging of temperature and species concentration, and is experiencing a surge of interest due to progress in laser technology, spectroscopy, and theoretical developments of nonlinear tomography techniques. The recent advancements in broad bandwidth, frequency-agile laser sources massively enrich the spectral information obtainable in TAS. Furthermore, nonlinear tomography enables the recovery of multiplexed information from a single tomographic inversion. The utilization of multispectral information improves the immunity of TAS to experimental noise and makes possible the simultaneous imaging of temperature, pressure, and multiple species. Nonlinear tomography can also be used to empower the imaging potential of sensitive and robust absorption techniques, such as wavelength modulation spectroscopy, for use in harsh and even optically dense environments. In combination, this greatly extends the applicability of TAS for more general and harsh scenarios in combustion technology. In this article we review basic concepts and mathematical foundations of classical absorption tomography, proceeding to more advanced recent concepts based on nonlinear tomography, and providing an extensive review of experimental demonstrations and practical applications in the context of state-of-the-art combustion research. (C) 2016 Elsevier Ltd. All rights reserved.