This is a comprehensive, critical, and pedagogical review of volumetric emission tomography for combustion processes. Many flames that are of interest to scientists and engineers are turbulent and thus inherently three-dimensional, especially in practical combustors, which often contain multiple interacting flames. Fortunately, combustion leads to the emission of light, both spontaneously and in response to laser-based stimulation. Therefore, images of a flame convey path-integrated information about the source of light, and a tomography algorithm can be used to reconstruct the spatial distribution of the light source, called emission tomography. In a carefully designed experiment, reconstructions can be post-processed using chemical kinetic, spectroscopic, and/or transport models to extract quantitative information. This information can be invaluable for benchmarking numerical solutions, and volumetric emission tomography is increasingly relied upon to paint a more complete picture of combustion than point, linear, or planar tools. Steady reductions in the cost of optical equipment and computing power, improvements in imaging technology, and advances in reconstruction algorithms have enabled a suite of three-dimensional sensors that are regularly used to characterize combustion. Four emission modalities are considered in this review: chemiluminescence, laser-induced fluorescence, passive incandescence, and laser-induced incandescence. The review covers the reconstruction algorithms, imaging models, camera calibration techniques, signal physics, instrumentation, and post-processing methods needed to conduct volumetric emission tomography and interpret the results. Limitations of each method are discussed and a survey of key applications is presented. The future of volumetric combustion diagnostics is considered, with special attention paid to the advent and promise of machine learning as well as spectrally-resolved volumetric measurement techniques.