We apply background-oriented schlieren (BOS) imaging with computed tomography to reconstruct the instantaneous refractive index field of a turbulent flame in 3D. In BOS tomography, a network of cameras are focused through a variable index medium (such as a flame) onto a background of patterned images. BOS data consist of pixel-wise "deflections" between a reference and distorted image, caused by variations in the refractive index along the path between the camera and background. Multiple simultaneous BOS images, each from a unique perspective, are combined with a tomography algorithm to reconstruct the refractive index distribution (or optical density) in the probe volume. This quantity identifies the edges of the wrinkled turbulent flame surface. This is the first application of BOS imaging to flame tomography, setting the stage for low-cost 3D flame thermometry. Tomography is carried out within the Bayesian framework, using Tikhonov and total variation (TV) priors. The TV prior is more compatible with the abrupt spatial variation in the refractive index field caused by the flame front. We demonstrate the suitability of TV regularization using a proof-of-concept simulation of BOS tomography on an LES phantom. The technique was then used to reconstruct the instantaneous 3D refractive index field of an unsteady natural gas/air flame from a Bunsen burner using a 23-camera setup. Our results show how BOS tomography can capture and visualize 3D features of a flame and provide benchmark data for simulations. (C) 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.