Bayesian methods are a popular research direction for inverse problems, but this framework has not achieved widespread use in tomographic imaging. Primarily, this is because traditional tomographic uncertainty estimators are computationally expensive, but also because there is no well established way to interpret tomographic uncertainty estimates. This thesis presents a Bayesian model for tomography, where the tomographic image is assumed to follow a mean-field Gaussian distribution. Using this assumption, an iterative reconstruction algorithm can be derived which produces two images, 1) a maximum a posteriori (MAP) estimator, which is equivalent to a typical tomographic image reconstruction, and 2) a no computational cost tomographic uncertainty estimate, where each pixel of the reconstruction is associated with an individual standard deviation or variance estimate. In practice, the algorithm is derived using stochastic variational inference, which motivated its name; variational tomography (VT). VT is demonstrated by application to full-waveform inversion (FWI). FWI is a computationally complex reconstruction algorithm because the tomographic data are simulated using a full numerical simulation of an acoustic wavefield. The mean-field Gaussian is a restrictive assumption, which has implications for the accuracy of the posterior. However, it is demonstrated that the MAP estimator is equivalent to a traditional iterative reconstruction and the no cost uncertainty estimate has some useful properties. Specifically, the tomographic uncertainty estimates are evaluated in-silico and in-vitro, qualitatively and quantitatively, for the purpose of image quality analysis (IQA). A comparison between different methods of uncertainty estimation is not provided because different uncertainty estimators make different assumptions; there is no reliable ground truth. Instead, this study demonstrates that VT uncertainty is a useful tool for no-reference IQA. Thus, VT does not alter the dynamics of a traditional iterative reconstruction algorithm, but provides an additional tomographic uncertainty estimator with no additional computational cost.